Continuum mechanics has always been a rich source of inspiration for many areas of analysis. The problems studied there are often highly non-trivial, yet by their very nature inherently geometrical, possessing natural estimates. At the same time many questions in continuum mechanics have been strongly linked to the calculus of variations, in particular the analysis of static and quasistatic problems in solid mechanics. Far less common, however, are variational results for dynamic problems involving inertia.
The aim of this talk is to present an overview over some recent results (obtained together with B. Benešová, S.Schwarzacher and many others), which extend previous considerations from quasistatic problems to those involving inertia. These allow us to tackle questions of existence even for highly non-linear problems such as those of fluid-structure interaction, where not even the domain of the equations is known a-priori. In addition to that, these methods also reveal a lot about the geometry of the problem, both in the physical space, as well as the abstract state-space.
At the same time this talk will serve as a high-level overview and an invitation to a corresponding class on "Variationelle Methoden in der Kontinuumsmechanik/Variational methods in continuum mechanics" during the second half of this semester. In this class we will go through these results and the required tools in more detail. Thus all interested students are explicitly invited to attend both.
In this talk, I will introduce my research and interests. The primary focus will be on the regularity theory of variational problems (possibly with degeneracy), which are connected to geometric measure theory in various ways. A significant part of the talk will be about variational problems under the quasi-convexity assumption. In particular, I will discuss quasi-convex variational problems involving constant rank operators, with one example being differential forms, and explain how to deal with the degeneracy of such operators. In addition, the unique continuation problem on locally flat domains will be briefly mentioned.
The generalized SQG family of equations interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation; a well-known model for the three-dimensional Euler equations. These equations can be realised as geodesic equations on groups of diffeomorphisms. I will present some results pertaining to Fredholmness of the corresponding exponential maps, as well as the distribution of conjugate points. This is joint work with Martin Bauer, Gerard Misiołek and Stephen Preston.
I will talk about the global existence/blow-up properties and the long-term behavior for the solutions of the scalar conservation law with viscosity $ u_t+(bu^{k+1})_x=u_{xx} $ in the case $ b $ belongs to the Lorentz space $ L^{p,\infty} $.
This is a joint work with Stefano Bianchini.
In the work of V. Arnold (1966), it was shown that the incompressible Euler equations can be viewed as the L2-geodesic equations on the Lie-group of volume preserving diffeomorphisms. This idea has since been used to study various fluid flows through their flow map. In this talk, I will present some recent works in computational fluid dynamics using the diffeomorphism group structure of the Euler equations. I will then propose some potential applications of these diffeomorphism-based numerical methods to the study of mixing enhancement and anomalous dissipation.
In the first part of the talk, I will present our work on the "characteristic mapping method", a numerical scheme for the Lie-advection equation based on computing the inverse Lagrangian flow map. This method has been applied to the linear advection and Euler equations in 2 and 3D. We achieve fully non-diffusive transport of advected quantities, where the numerical solutions are not truncated in frequency space.
I will present in the second part of the talk potential new numerical tools that could assist in the study of mixing enhancement and anomalous diffusion. In particular, using the advection flow map, the advection-diffusion equation can be pulled-back to Lagrangian frame to an anisotropic diffusion equation whose anisotropy depends on the pullback metric.
The talk will address the local regularity of axisymmetric suitable weak solutions, emphasising the possibility of Type II blowups only. Additionally, the discussion will touch upon the regularity of solutions with slightly supercritical restrictions.
In this talk, I present a new technique to prove two-dimensional crystallization results in the square lattice for finite particle systems. We consider configurational energies featuring two-body short-ranged particle interactions and three-body angular potentials favoring bond-angles of the square lattice. To each configuration, we associate its bond graph which is then suitably modified by identifying chains of successive atoms. This method, called stratification, reduces the crystallization problem to a simple minimization that corresponds to a proof via slicing of the isoperimetric inequality.
We present a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the current distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimization problems up to degrees of freedom given by rotations: one for the evolution of mean and covariance of the interpolating curve, and one for its shape. Similarly, on the level of the gradient flows a similar splitting into the evolution of moments and shapes of the distribution can be observed. Those show better convergence properties in comparison to the classical Wasserstein metric in terms of exponential convergence rates independent of the Gaussian target.
In this seminar we outline an example of a divergence free velocity field u ∈ Cα([0, 1] × T2), with α < 1, for which there is lack of selection of solutions to the transport equation via vanishing viscosity. With the same construction of the velocity field, rescaled in time and enjoying different regularity, we prove anomalous dissipation in the full Obukhov–Corssin supercritical regime for the advection diffusion equation. If the time permits we will also mention some new results about anomalous dissipation for the forced Navier–Stokes equations.
These are joint works with Elia Bru´e, Maria Colombo, Gianluca Crippa and Camillo De Lellis.
I will speak about several models of interacting particle systems (such as Asymmetric Simple Exclusion Process) and their limits, in which various Partial Differential Equations arise.
We consider the incompressible porous media equation (IPM) describing the evolution of an incompressible fluid in a porous medium subject to gravity. The initial data of our interest consists of a (not necessarily flat) interface separating a heavier fluid with homogeneous density $\rho_+>0$ from a lighter fluid with homogeneous density $\rho_-\in(0,\rho_+)$, with the heavier one being above the lighter one. Due to the gravity term this situation is in real world scenarios unstable and mathematically ill-posed as an initial value problem. The talk addresses the question of recovering well-posedness on the level of averaged solutions (convex integration subsolutions) by means of a selection based on maximal potential energy dissipation. We will see that this criterion leads to a nonlocal hyperbolic conservation law - the "macroscopic IPM" system - which is consistent with the relaxation of Otto based on the gradient flow structure of IPM. In the second part of the talk we will discuss the construction of an entropy solution for macroscopic IPM emanating from a real analytic initial interface.This is based on a joint work with Angel Castro and Daniel Faraco.
In this talk we will introduce Bressan's Fire Conjecture: it is concerned with the model of wild fire spreading in a region of the plane and the possibility to block it using barriers constructed in real time.
The fire starts spreading at time $t=0$ from the unit ball $B_1(0)$ in every direction with speed $1$, while the length of the barrier constructed within the time $t$ has to be lower than $\sigma t$, where $\sigma>0$ is a positive constant (construction speed). In 2007 Bressan conjectured that if $\sigma\in[1,2]$ no barrier can block the spreading of the fire. In this talk we will prove Bressan's Fire Conjecture in the case barriers are spirals. Spirals are thought to be the best strategies a firefighter can do in order to confine the fire for $\sigma\leq 2$. We will introduce the new concept of family of generalized barriers and we will prove that, if there exists such a family satisfying a diverging condition, then no spiral can confine the fire. This is a joint work with Stefano Bianchini.
In this talk I will explain how to obtain precise informations on the structure of the free-boundary to $2$-dimensional solutions of the one and two phase problems at so-called branching points using the theory of (quasi-)conformal maps. The talk is based on joint work with G. De Philippis (Courant) and B. Velichkov (Pisa).
The Burkholder function, found in 1984 in the context of sharp martingale inequalities, turns out to be a “master function” which rules the sharp integrability properties of mappings in the plane. In this talk we will describe a proof of what we call the Burkholder area inequality, which is an optimal estimate for the Burkholder energy of quasiconformal maps in the spirit of the classical Grönwall-Bieberbach area formula. The talk is based on joint work with K. Astala, D. Faraco, A. Koski and J. Kristensen.
The Cwikel-Lieb-Rozenblum (CLR) inequality is a semi-classical bound on the number of bound states for Schrödinger operators. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel’s proof is said to give a constant which is at least about 2 orders of magnitude off the truth.
In this talk I will give a brief overview of the CLR inequality and present a substantial refinement of Cwikel’s original approach which leads to an astonishingly good bound for the constant in the CLR inequality. Our proof is quite flexible and leads to rather precise bounds for a large class of Schrödinger-type operators with generalized kinetic energies. Moreover, it highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis. (joint work with D. Hundertmark, P. Kunstmann, and S. Vugalter)
When Kerr black holes rotate at their maximally allowed angular velocity, they are said to be extremal. Extremal black holes are critical solutions to Einstein’s equations of general relativity. They exhibit interesting phenomena that are not present in more slowly rotating black holes. I will introduce recent work on the existence of strong asymptotic instabilities of a non-axisymmetric nature for scalar waves propagating on extremal Kerr black hole backgrounds and I will discuss the connection with previously known axisymmetric instabilities as well as with late-time power law tails in gravitational radiation.
In this talk we consider the evolution of an interface evolving by an incompressible flow. On the one hand, we study the one-phase Muskat problem, where the fluid is filtered in a porous medium. In the gravity-stable case, we show that initial Lipschitz graphs of arbitrary size provide global-in-time well-posedness. On the other hand, we study the interface dynamics given by two fluids of different densities evolving by the linear Stokes law. We show stability to the flat stable case and exponential growth in the unstable regime.
In this talk we will review some results on rigorous a posteriori error estimates for numerical approximations of systems of hyperbolic conservation laws, i.e. bounds for discretization errors that can be computed from numerical solutions without making assumption of the properties of the exact solution. We will explain the fundamental link between a posterirori error estimates and stability properties of the PDE that is to be approximated.
We will describe a posteriori error estimates that have been derived a few years ago based on relative entropy stability estimates. We will outline recent progress in a posteriori error estimates for one-dimensional hyperbolic conservation laws based on two approaches: Firstly, results using Bressan's stability theory and, secondly, results using a-contraction estimates based on work of Vasseur and Krupa.
Quantitative rigidity results, besides their inherent geometric interest, have played a prominent role in the mathematical study of variational models related to elasticity\plasticity. For instance, the celebrated rigidity estimate of Friesecke, James, and Müller has been widely used in problems related to linearization, discrete-to-continuum or dimension-reduction issues for energies within the framework of nonlinear elasticity. After a quick review of the aforementioned results, we will present appropriate generalizations to the setting of variable domains, where the geometry of the domain comes into play in terms of a suitable surface energy of its boundary. If time permits, we will also discuss applications of this new rigidity estimates in questions related to dimension reduction for elastic materials with free surfaces. This is joint work with Manuel Friedrich and Leonard Kreutz.
In continuum physics the underlying laws have to be satisfied not only for an entire body but also for all of its subbodies. Classically these laws account for contact interactions between contiguous subbodies exerted across their common boundary and a well known result by Cauchy constitutes that contact interactions having a continuous density only depend linearly on the normal field of the common boundary. Recent extensions of this theory that also cover the occurrence of certain concentrated contact interactions showed that Cauchy's fundamental result remains true for a suitable collection of sets of finite perimeter as common boundaries of subbodies. But at the same time the classical approach results in a number of problems, among other things, when concentrations occur.
We will present a new approach by Schuricht, which retains the advantages of the classical theory, but at the same time solves some of its problems and allows a more precise description of concentrations and if time permits also sketch possible extensions of this approach. This is joined work with Friedemann Schuricht.
We present a dimension-incremental algorithm for approximating high-dimensional, multivariate functions in an arbitrary bounded orthonormal product basis. The goal is a truncation of the basis expansion of the function, where the corresponding significant index set is not known in advance. Our method is based on point evaluations of the considered function and adaptively builds a reasonable index set, such that the approximately largest basis coefficients are included. We provide an proof idea of a detection guarantee for such an index set under certain assumptions on the sub-methods used within our algorithm, which can be used as a foundation for similar statements in various other situations as well.
The application of a modification of the developed adaptive method to partial differential equations which depend on several random parameters allows for a numerical classification of important and non-important random parameters as well as significantly interacting parameters.
Abstract: In this talk we construct solutions to Ricci de Turck flow in four dimensions on compact manifolds which are instantaneously smooth but whose initial values are (possibly) non-smooth Riemannian metrics whose components, in smooth coordinates, belong to $W^{2,2}$ and are bounded from above and below. A Ricci flow related solution is constructed whose initial value is isometric in a weak sense to the initial value of the Ricci de Turck solution. (joint work with Tobias Lamm)
In this talk we give an introduction to deterministic and stochastic models of epidemics. We are interested in epidemics evolving on social networks, especially in the limit behaviour when the size of the underlying graph goes to infinity. For this we describe some notions of graph limits and a recent result describing the PDE limit of the stochastic process on the "not too sparse" regime where the average degrees are diverging but might be much smaller compared to the number of vertices.
In the second part, we are investigating a property of the PDE limit dubbed the Universality of Small Initial Conditions (USIC). We show that if initially the ratio of infected individuals is small than any initial configuration will lead to approximately the same epidemic curve up to time translation. The limit curve starting from the infinite past from infinitesimally small infections is identified as the "nontrivial eternal solution" of the PDE, well defined for both all positive and negative time values and connecting the infection less constant zero solution at minus infinity with the stationary endemic solution at plus infinity.
A nonlinear wave equation with a double-well potential in 1+1 dimension admits stationary solutions called kinks and antikinks, which are minimal energy solutions connecting the two minima of the potential. We study solutions whose energy is equal to twice the energy of a kink, which is the threshold energy for a formation of a kink-antikink pair. We prove that, up to translations in space and time, there is exactly one kink-antikink pair having this threshold energy. I will explain the main ingredients of the proof. Joint work with Michał Kowalczyk and Andrew Lawrie. One day before the seminar, an announcement with the link will be sent to the mailing list of the groups of Prof. Otto and Gess. If you are not on the mailing list, but still want to join the broadcast, please contact Katja Heid.
In this talk I will consider the density patch problem for the two dimensional inhomogeneous incompressible viscous flow. I will show that the Besov regularities as well as the tangential regularities of any order for the velocity field can be propagated globally in time. The proof relies on the (time weighted) energy estimates. This is a joint work with Ping Zhang from Beijing.
Quasiconvexity and rank-one convexity play a fundamental role in the vectorial Calculus of Variations; Morrey’s problem is to decide whether these notions coincide. In this talk we will see that to solve this problem it is sufficient to check whether all extremal rank-one convex functions are quasiconvex. We will also identify a class of such extremal functions, thus proving a conjecture made by Šverák in 1992.
I will discuss the application of the method of compensated compactness to the compressible, isentropic Euler equations under certain geometric assumptions, e.g. the case of fluid flow in a nozzle of varying cross-sectional area or the assumption of planar symmetry under special relativity. Under these assumptions, the equations reduce to the classical (or relativistic) one-dimensional isentropic Euler equations with additional geometric source terms. In this talk, I will explain how the classical strategy of DiPerna, Chen et. al. can be adapted to handle these more complicated systems and will highlight some of the difficulties involved in extending the techniques to the relativistic setting.
In recent years, following the seminal works of Villani and Mouhot on Landau damping, phase-mixing as a damping mechanism and inviscid damping in fluid dynamics have attracted much interest. Here, mixing in physical space and weak convergence interact with the Biot-Savart law and results in damping of the velocity field.
In this talk, I will discuss linear stability and damping with optimal decay rates around 2D Taylor-Couette flow between two concentric cylinders and similar circular flows. A particular focus will be on asymptotic singularity formation at the boundary and stability in weighted spaces.
In the theory of transport equation arises a problem of representation of a distribution div(β(u)b) where u is a scalar field, b is a vector field with bounded variation and β is a smooth function. Such representation has been obtained by L. Ambrosio, C. De Lellis and J. Maly, up to a singular measure with particular properties. The authors have also obtained a sufficient condition when this measure is zero. In connection with the Bressan's compactness conjecture they have posed a question of existence of a nearly incompressible vector field for which this sufficient condition is not satisfied. An example of such vector field, which has been recently constructed, will be presented.
In this talk we will discuss a counterexample to the well-posedness of entropy solutions to the Cauchy problem for the compressible Euler equations in two space dimensions. In particular we show the existence of classical Riemann data, i.e. pure jump discontinuities across a line, for which there are infinitely many entropy weak solutions (bounded away from the void). We also show that some of these Riemann data are generated by a 1-dimensional compression wave: our theorem leads therefore to Lipschitz initial data for which there are infinitely many global bounded entropy weak solutions. Our methods rely heavily on the new analysis of the incompressible Euler equations recently carried out by De Lellis and Székelyhidi and based on a revisited "h-principle".
I will present a regularity result for degenerate elliptic equations in nondivergence form. In joint work with Charlie Smart, we prove a Harnack inequality for equations with possibly unbounded ellipticity-- provided that the ellipticity belongs to $L^d$ (where $d$ is the dimension). As an application we obtain a stochastic homogenization result for such equations (and a new estimate for the effective coefficients) as well as an invariance principle for random diffusions in random environments.
I will discuss nonlocal integral continuity equations for which notrivial stationary states show interesting properties. We will discuss their nonlinear stability/instability and the dimension of their support. They lead to challenging questions of calculus of variations in measure settings.
Many-particle storage systems can be modelled by a nonlocal Fokker-Planck equation that describes energy minimisation in a double well-potential, involves two small parameters and is driven by a time-dependent constraint. The dynamics takes place on three different time scales and can produce several types of hysteretic or non-hysterectic phase transitions. Formal asymptotic analysis allows to derive reduced dynamical models for different small-parameter limits. In this talk we in particular discuss the fast-reaction regime that is governed by Kramer's formula, and also indicate how the limit procedure can be made rigorous
This is joint work with Michael Herrmann and Juan Velazquez.
Consider the heat equation with random coefficients on Z^d. The randomness of the coefficients models the inhomogeneous nature of the medium where heat propagates. We assume that the distribution of these coefficients is invariant under spatial translations, and has a finite range of dependence. It is known that if a solution to this equation is rescaled diffusively, then it converges to the solution of a heat equation with constant coefficients. I will first explain how this convergence can be rephrased in terms of the long-time behaviour of an associated random walk. Based on this representation, I will then present recent progress on the estimation of the speed of this convergence.
The Heston stochastic volatility process is a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerate-elliptic partial differential operator, where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half-plane. In mathematical finance, solutions to obstacle problem for the elliptic Heston operator correspond to value functions for perpetual American-style options on the underlying asset. With the aid of weighted Sobolev spaces and weighted Hölder spaces, we establish the optimal C^{1,1} regularity (up to the boundary of the half-plane) for solutions to obstacle problems for the elliptic Heston operator when the obstacle functions are sufficiently smooth. This is joint work with Panagiota Daskalopoulos. While regularity theory for obstacle problems can be a technical area, our aim is to give a presentation which is accessible to PhD students.
Preprint: http://arxiv.org/abs/1206.0831
I will discuss the asymptotic behavior of perturbations of transition front solutions arising in Cahn--Hilliard systems on $\mathbb{R}$. Such equations arise naturally in the study of phase separation processes, and systems describe cases in which three or more phases are possible. When a Cahn--Hilliard system is linearized about a transition front solution, the linearized operator has an eigenvalue at 0 (due to shift invariance), which is not separated from essential spectrum. In many cases, it's possible to verify that the remaining spectrum lies on the negative real axis, so that stability is entirely determined by the nature of this leading eigenvalue. I will discuss the nature of this leading eigenvalue and also the verification that spectral stability---defined in terms of an appropriate Evans function---implies phase-asymptotic stability.
In this talk, I would like to introduce the class of weakly stationary-harmonic multiple-valued functions in the Sobolev space of multiple-valued functions. This class of functions are defined as the critical points of Dirichlet integral under smooth domain-variations and range-variations.
I would show the interior continuity for this class of functions as the domain-dimension is two.
The problem of estimating entropy numbers of convex hulls has attracted researchers in geometrical functional analysis over the last 30 years. At first, we tend to shed some light on the tight relations between linear operators, Gaussian stochastic processes and convex hulls. In a second step, we will show how to treat the analytical task to estimate the entropy numbers of convex hulls in Hilbert space using a probabilistic approach. In particular, a precise link between the convex hull and the reproducing kernel Hilbert space of a Gaussian process enables us to employ results on small ball probabilities. Vice versa, we show how small ball probability estimates can be obtained from the entropy of convex hulls.
In this talk, we study a class of scalar, parabolic, semi-linear stochastic partial differential equations perturbed by a space-time white noise on a bounded real intervalin the small noise asymptotic.
Due to the stochastic term, metastable transitions occur between different stable equilibriums of the deterministic dynamical system and different behaviors on different timescales happen. We compute the expectation of the transition times for some models (so-called Eyring-Kramers Formula).
The proof use a finite difference approximation and a coupling and apply finite dimensional estimates to the approximation (using potential theory and capacities). We prove the uniformity of the estimates in the dimension and then we take the limit to recover the infinite dimensional equation.
In this talk I will present a unified approach for the effect of fast rotation and dispersion as an averaging mechanism for, on the one hand, regularizing and stabilizing certain evolution equations. On the other hand, I will also present some results on which large dispersion is a destabilizing mechanism for the long-time dynamics of certain dissipative evolution equations.
The emergence of structure from disorder is interesting in several physical models such as vortex coalescence in 2d flows, and domain growth in materials science. One mathematical model for such phenomena is to study the equations of continuum physics with random initial data.
We show that this problem has a very rich structure even for the simplest nonlinear equations. Our main result is that the evolution of shock statistics for scalar conservation laws with convex flux and suitable random data is completely integrable. These results sit at an interesting junction of kinetic theory, integrable systems, and probability theory. Little background will be presumed in any of these areas.
In this talk, we study the fragmentation of a discrete d-dimensional torus (d>=3) by a simple random walk, a basic mathematical model for the gel degradation by an enzyme. We focus on percolative properties of the largest and the second largest connected components in the complement of the trace of the random walk, as time evolves. We describe time scales on which macroscopic structural changes occur. Our analysis is based on a connection between the microscopic structure of the random walk trace in the bulk and the so-called random interlacements. This is a joint project with A. Drewitz and B. Rath (both from ETH Zurich).
We study couplings $q^\omega$ of the Lebesgue measure $\frak L^d$ and the Poisson point process $\mu^\omega$ on $\mathbb R^d$. We ask for a minimizer of the mean $L^p$-transportation cost.
The minimal mean $L^p$-transportation cost turns out to be finite for all $p\in (0,\infty)$ provided $d\ge3$. If $d\le2$ then it is finite if and only if $p
We shall focus on the higher integrability property enjoyed by the approximate gradients of local minimizers of the 2d Mumford-Shah energy. Related regularity issues will be also discussed.
This is joint work with C. De Lellis (Universitaet Zuerich).
In the deterministic regularity theory for parabolic systems, there are both classes of systems, where all weak solutions are actually more regular, and examples of systems which admit solutions that develop singularities in finite time. In this talk we investigate the regularity of solutions under the effect of random perturbations. On the one hand, we present an extension of a regularity result (due to Koshelev and Kalita) to the stochastic setting. On the other hand we discuss a partial result that might allow to attack the (open) question whether or not stochastic noise might even prevent the emergence of singularities for some particular systems.
(Joint work with F. Flandoli, Pisa)
This talk concerns integral varifolds (i.e. generalised submanifolds) of arbitrary dimension and codimension in an open subset of Euclidean space with its first variation of area given by either a Radon measure or a function in some Lebesgue space. In most cases, optimal pointwise decay results for the quadratic tilt-excess (i.e. the mean oscillation of the tangent planes measured in L^2) are established for those varifolds.
The viewpoint of the talk will be that of elliptic partial differential equations.
We consider the spin-coating process which is described by the Navier-Stokes equations in a layer-like domain in $\mathbb{R}^3$ in the rotating setting. Our model takes into account Coriolis forces, centrifugal Navier-Stokes equations in a layer-like domain in $\mathbb{R}^3$ in the rotating setting. Our model takes into account Coriolis forces, centrifugal forces as well as surface tension on the free boundary. On the fixed boundary we prescribe Robin boundary conditions. Our aim is to show local existence and uniqueness of strong solutions. In order to do so, we transform this problem to a fixed layer by the Hanzawa transform and show maximal regularity estimates for a suitable linearized problem. We also present some extension to the non-Newtonian case.
We prove a bubble-neck decomposition and an energy quantization result for sequences of Willmore surfaces immersed into R^m (m>2) with uniformly bounded energy and non-degenerating conformal type. We deduce the strong compactness (modulo the action of the Moebius group) of closed Willmore surfaces of a given genus below some energy threshold.
This is joint-work with Tristan Rivière (ETH Zürich).
In the first part of the seminar I will overview some results concerning one-dimensional systems of conservation laws and viscous approximations of initial-boundary value problems. The main challenges posed by the initial-boundary value problem are the possible presence of boundary layer phenomena and the fact that, in general, the limits of two different viscous approximation do not coincide.
In the second part of the seminar I will focus on results, obtained in collaboration with S. Bianchini and C. Christoforou, concerning the limit of a viscous approximation in the case of the so-called boundary Riemann problem. In particular, I will describe a possible characterization of the limit and I I will establish a uniqueness criterium.
In Collaboration with Makiko Sasada (Keio University, Japan).
We consider a chain of an-harmonic oscillators whose Hamiltonian dynamics is perturbed by a local energy conserving noise. Under a diffusive space-time rescaling, we prove that the fluctuations of the energy evolve following a linear SPDE, with diffusion coefficient given by the Green-Kubo formula.
Understanding the processes of Life requires all tools and toys available for us as natural scientists -- and mathematicians. As such we have learned to play with elementary ingredients and their suitable combinations, in order to invent dynamical systems and make them work and function in a proper and useful way. What else can we see in Nature, where on the evolutionary time scale functionable and useful dynamical systems have been invented and properly developed during millions of cell generations.
Here we consider, as prototypical example, the ubiquitous phenomenon of crawling cell motility, with the reactive and contractile cytoplasm system as elementary ingredient in combination with associated crosslinker and plasma membrane proteins. Their mechano-chemical properties and mechanisms span a large range of spatio-temporal scales, from molecular events as polymerization or adhesion receptor binding (nm; sec) over cytoplasmic flow and cell deformation (µm; min) up to effective cell translocation and path generation (mm; h). Since an important task of modelling is to reconstruct, simulate and understand the essential mechanisms behind the observed modes of Life, the invented mathematical models have to be treatable and accessible for further analysis.
Therefore, in this talk we present simplified 1-D approximations of a general 2-D hyperbolic-elliptic-parabolic stochastic PDE system with free boundary conditions describing the dynamics of single cell adhesion and motility. The basic problem is to reproduce the observed transitions between a non-polarized cell with on-spot motility and a polarized cell with directionally persistent cell locomotion.
This talk will describe some physical phenomena that occur when a solid is subject to a stress that greatly exceeds its yield stress and also discuss a mathematical model for uniaxial wave motion in such a solid. This model leads to some interesting questions concerning degeneracy in hyperbolic pdes and causality of interfaces.
Solving flow and transport equations on heterogeneous multiscale porous media is a numerically demanding task. One may either reduce the complexity of the model by scaling the equations and transfer the model to large scales with a coarser resolution or improve the numerical solvers towards large scale solvers.
Ensemble averaging, volume averaging and two scale expansions are scaling methods which have been developed and successfully applied to scale up flow and transport processes in heterogeneous media. Tremendous work has been devoted to give explicit results for effective or larger scale parameters making use of perturbation theory approaches. Numerical multiscale schemes have been developedwhich step beyond perturbation theory approximations. Limitations of existing approaches are often not handled carefully which may lead to unphysical averaging effects like for example overestimated mxing in solute transport or wrong hydraulich heads in pumping tests.
In this talk, a scaling method will be introduced which is coarsening the equations in infinitesimal small coarsening steps and averaging stepwise subscale effects (Coarse Graining). Imposing the requirement of flux conservation across scales, a differential equation for the scale dependence of effective parameters in flow and transport is derived. These type of equations are well known in theoretical physics where they are called renormalisation group equations. Knowing the scale dependence of effective parameters, transient and preasymptotic effects of flow and transport through hetereogeneous media can be described more realistically avoiding unphysical averaging effects. I will present results for scale dependent effective parameters for various flow and transport problems.
The results might be also important for efficient large scale solvers like multigrid solvers. Multigrid solvers are based on restriction and interpolation schemes between coarser and finer grids and the construction of coarse grid operators is central. Knapek hypothesized that coarse grid operators in multigrids should be physically meaningful (SIAM J. Sci. Comput ,1996/) /in order to achieve good mathematical accuracy and numerical convergence rates. We tested this hypothesis for the ellliptic equation with oszillating parameters.
For many classical rigidity questions in differential geometry it is natural to ask to which extent they are stable. I will review several results in the literature and mention a few applications.
As a model for layered nanomagnets with applied current we take Landau-Lifschitz-Gilbert equations with a spin torque term. We study the case of a aligned external field and material anisotropy in one space dimension. The resulting symmetry allows to rather easily prove existence of a large class of coherent structures, which are patterns built up from wavetrains or constant states in analogy to those studied in complex Ginzburg-Landau equations. Moreover, the spectral stability of wavetrains can be completely characterized; wavetrains can lose stability only through a sideband instability. This is joint work with Christoph Melcher (RWTH).
A deterministic coalescing dynamics with constant rate for a particle system in a finite volume with a fixed initial number of particles is considered. It is shown that, in the thermodynamic limit, with the constraint of fixed density, the corresponding coagulation equation is recovered and global in time propagation of chaos holds.
Assessing and hedging a future uncertain liability is one of the central questions of finance. In 1999, Artzner, Delbaen, Eber and Heath provided an axiomatic approach which gave the definition of "coherent risk measures". Later Foellmer and Schied, relaxed the positive homogeniety to define the "convex risk measures".
Mathematically, these are maps, with certain properties, which assign a numeric value to each random variable representing the future liability. One may think of them as the capital requirement for carrying the financial positions. Also, they turn out to be nothing but classical utility functions (up to a sign convention) with the additional property called "cash-invariance". In the Markovian situations, these are closely related to nonlinear, scalar, parabolic partial equations. Under the assumptions of the early studies, the resulting equations are semilinear. Recently, this assumptions have been relaxed to allow convex but fully nonlinear equations.
In this talk, I will outline the theory of risk measures, backward stochastic differential equations and their connections to PDEs. This is joint work with Nizar TOUZI (Ecole Polytechnique, Paris) and Jianfeng ZHANG (UCS).
I will talk about some recent developments on the interplay between the L^2 and W_2 geometries. Results include the possibility of studying the continuity equation in a genuine metric framework, and the identification of the heat flow as gradient flow in spaces with Ricci curvature bounded from below.
There is strong evidence of asymptotically self similar blow up for the super critical nonlinear Schroedinger equation and the related Korteweg-de-Vries equation. Recently Merle, Raphael and Szeftel studied the blow-up dynamics for the slightly supercritical nonlinear Schrödinger equation.
In this talk I will present a bifurcation analysis for the bifurcation of self similar finite energy solutions from the travelling wave solution for the focussing generalized Korteweg-de-Vries equation. The difficulty comes from a rich family of asymptotic regimes, which have to be understood and controlled simultaneously.
In this talk we will show how the mechanical response of an elastic film is affected by subtle geometric properties of its mid-surface. The crucial role here is played by spaces of weakly regular (Sobolev) isometries or infinitesimal isometries. These are the deformations of the mid-surface preserving its metric up to a certain prescribed order of magnitude, and hence contributing to the stretching energy of the film at a level corresponding to the magnitude of the given external force.
In this line, we will discuss results concerning the matching and density of infinitesimal isometries on convex, developable and axisymmetric surfaces. By a matching property, we refer to the possibility of modifying an infinitesimal isometry of a certain order to make it an infinitesimal isometry of a higher order. In particular, we will show that on a convex surface, any one parameter family of first order bendings generated by a first order isometry can be modified at a higher order of perturbation to a family of exact bendings (exact isometries).
In the second step, we will show how this analysis can be combined with the tools of calculus of variations towards the rigorous derivation of a hierarchy of thin shell theories. The validity of each theory depends on the scaling of the applied force in terms of the vanishing thickness of the reference shell. The obtained hierarchy extends the seminal result of Friesecke, James and Muller valid for flat (plate-like) films, to shells whose mid-surface may have arbitrary geometry. When a matching property is established, the above-mentioned infinte hierarchy effectively collapses to a finite number of theories.
The G-equation is a nonlinear, level-set PDE that is used in models of turbulent combustion. Level sets of the solution represent a flame surface which moves with normal velocity that is the sum of the laminar flame velocity and the fluid velocity. I will present recent work on the large-time asymptotics of the solutions when the fluid velocity is given by a prescribed, nearly incompressible vector field which is either periodic or random and stationary. The main analytical challenge comes from the fact that the Hamiltonian in the equation is not coercive. This includes joint work with Pierre Cardaliaguet, Takis Souganidis, and Alexei Novikov.
I will describe joint work with Matt Elsey and Peter Smereka on large scale simulations of mean curvature and related geometric motions for networks of curves in 2D and surfaces in 3D. These motions arise as gradient flow for variational models encountered in image processing (Mumford-Shah model of image segmentation) and materials science (grain boundary motion in polycrystals). Our methods are motivated by an old idea due to Merriman, Bence, and Osher (diffusion generated motion), but are far more accurate on uniform grids.
In this talk I discuss work with Marco Veneroni on curvature-penalizing energies that arise in models of pattern formation. These nonlocal energies contain competing terms, penalizing both rapid variation and large-scale aggregation, and this competition results in structures with a preferred length scale.
I will concentrate on a model that produces striped patterns that may be curved and may show defects. One of the central challenges in the field of pattern formation is to rigorously connect the properties of the microscopic model on one hand with large-scale features of the striped pattern (such as curvature and defects) on the other. For the model at hand we have proved a result of this type, which connects the value of the microscopic energy to the presence or absense of certain large-scale features.
An interesting consequence of this result is the appearance of a new formulation of the eikonal equation, in terms of projection matrices. This formulation avoids unphysical singularities that arise in the usual vectorial formulation of the eikonal equation and raises many questions about the properties of this equation.
I shall introduce the simplest 3D model capable of reproducing the complex thermo-magneto-mechanical behavior of a magnetic shape memory alloy crystal featuring a cubic to tetragonal transformation (such as in Ni2MnGa). The model is based on a suitable extension of generalized plasticity. As such its existence will be analyzed within the energetic solvability frame. This is joint work with F. Auricchio, A.-L. Bessoud, M. Kruzik, and A. Reali.
Bayessche Schätzverfahren haben in Natur- und Ingenieurwissenschaften mit wachsender Verfügbarkeit schneller und konstengünstiger Rechenleistung seit den 1980er Jahren an Bedeutung gewonnen. Die mathematische Analyse dieser Verfahren und das damit einhergehende profunde Verständnis hinkt dem tatsächlichen Stand der algorithmischen Entwicklung jedoch in mancherlei Hinsicht hinterher.
Im Vortrag wird anhand der Grundaufgabe der Signalverarbeitung gezeigt, wie mit Methoden der stochastischen Analysis Stabilität und Sensitivität einer L2-optimalen Schätzung eines Markovschen Signals qualitativ und quantitativ verstanden werden kann. Die Anwendungsbreite reicht dabei von klassischen Filtern in Ingenieuranwendungen aus dem Bereich der Steuerungs- und Regelungstechnik ueber statistische Inferenz stochastischer Prozesse bis hin zu Bayesschen Inversen Problemen. Stochastische Analysis unterstützt auch die Entwicklung und theoretische Analyse stochastischer Algorithmen zur numerischen Berechnung der optimalen Schätzung, die im Vortrag ebenfalls kurz besprochen werden sollen.
A droplet spreading on a glass, in spite of being a very common phenomenon, raises quite a few problems of basic nature whose solution is yet debated: among them, the modeling of the interface which separates "dry" and "wet" regions. The simplest way to explore these issues is to look at the lubrication regime, in which droplets may be modeled by free boundary problems for fourth-order degenerate PDEs, the so-called thin-film equations (TFEs). After briefly reviewing the state of the art and the main open problems for the TFE, I will concentrate on a regularity theory which has been introduced together with Hans Knuepfer and Felix Otto in the case of a linearly degenerate mobility and a zero contact-angle at the free boundary. The TFE is viewed as a classical free boundary problem, and the strategy is based on a-priori energy-type estimates which provide "minimal" conditions on the initial datum under which a unique, global, and smooth solution exists (in case of perturbations of the stationary solution).
Motivated by the study of Almgren's center manifold in the regularity of higher codimension mass-minimizing currents, I will present a new proof of the higher regularity ($C^{3, \alpha}$) of multiplicity one minimal surfaces without using the non-parametric Schauder's estimates. This is a joint work with C. De Lellis.
We consider the Relativistic Vlasov-Maxwell system of equations which describes the evolution of a collisionless relativistic plasma. We show that under rather general conditions, one can test for linear instability by checking the spectral properties of Schrodinger-type operators that act only on the spatial variable, not the full phase space. This extends previous results that show linear and nonlinear stability and instability in more restrictive settings.
Buckling of beams and plates is reasonably well-understood. In my talk I will emphasize the universality of buckling, opening the possibility of the common approach to buckling of bodies with complex geometry. Buckling is exhibited as a failure of non-negativity of second variation in non-linear elasticity. The universality of the frame indifference of energy densities for all hyper-elastic materials is seen to be the common cause of both flip and buckling instabilities under compressive loading. The constitutive linearization procedure is introduced, that takes into account both linear material response and the geometric non-linearity associated with frame indifference. Finally, the critical buckling load is understood mathematically as a generalized Korn constant in the Korn-type inequality.
Electronic structure models like density functional theory have been widely used in various areas. In this talk, we will discuss some recent progress in the mathematical study of these models. Emphasis will be put on the derivation of macroscopic continuum theory (nonlinear elasticity, macroscopic Maxwell equations, etc) from density functional theory.
In this talk I will present a new model for a specific infinite quantum system: a crystal with a defect.
The main idea is to describe at the same time the electrons bound by the defect and the (nonlinear) behavior of the infinite crystal. This leads to a (rather peculiar) bounded-below nonlinear functional whose variable is however an operator of infinite-rank.
I will provide the correct functional setting for this functional and discuss the properties of bound states. I will in particular relate them to the dielectric properties of the crystal. Finally, I will discuss discretization issues and show preliminary numerical results in 1D.
This is a review of joint works with Eric Cancès and Amélie Deleurence (Ecole des Ponts, Paris).
We consider the evolution of an interface, modeled by a parabolic equation, in a random environment. The randomness is given by a distribution of smooth obstacles of random strength. To provide a barrier for the moving interface, we construct a positive, steady state supersolution. This construction depends on the existence, after rescaling, of a Lipschitz hypersurface separating the domain into a top and a bottom part, consisting of boxes that contain at least one obstacle of sufficient strength. We prove this percolation result.
Furthermore, we examine the question of existence of a solution propagating with positive velocity in a random field with non-bounded random obstacle strength.
This work shows the emergence of a rate independent hysteresis in systems subject to a viscous microscopic evolution law through the interaction with a random environment.
Joint work with N. Dirr (Bath University) and M. Scheutzow (TU Berlin).
I will present the interest of using entropy techniques to study some techniques used in molecular dynamics for (i) sampling and (ii) building effective dynamics.
The aim of this talk is to study energy functionals concentrated on the jump set of 2D vector fields of unit length and of vanishing divergence. The motivation comes from thin-film micromagnetics where these functionals correspond to limiting wall-energies. The main issue consists in characterizing the wall-energy density (the cost function) so that the energy functional is lower semicontinuous (l.s.c.) in a certain space. The key point resides in the concept of entropies due to the scalar conservation law implied by our vector fields. Our main result identifies appropriate cost functions associated to certain sets of entropies. In particular, certain power cost functions lead to l.s.c. energy functionals. A second issue concerns the existence of minimizers of such energy functionals that we prove via a compactness result. A natural question is whether the viscosity solution is a minimizing configuration. We show that in general it is not the case for nonconvex domains. However, the case of convex domains is still open.
The talk is a joint work with Benoit Merlet, Ecole Polytechnique (Paris).
In a recent work, Felix Otto and the author have introduced and analyzed a numerical method to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. We have shown that the overall error is the sum of three terms: a stochastic error of variance type, a systematic error related to some regularizing parameter, and an error related to the approximation on a finite box. The stochastic error, which has the scaling of the central limit theorem, is the dominant error up to dimension 8. Then, the systematic error saturates and dominates the overall error --- this analysis is optimal. In this talk, I will quickly recall the framework and the results above, and then introduce a general class of formulas for the approximation of homogenized coefficients. These formulas have the advantage that the associated systematic error can be made of higher order than the stochastic error in any dimension, thus generalizing the results by Felix Otto and the author to d>8.
This is joint work with Jean-Christophe Mourrat, Université de Provence.
Stationary surfaces in Lorentzian space may arise as singular limits of the hyperbolic analog of Ginzburg-Landau equations. We describe some qualitative properties of such surfaces, assuming enough regularity. However, singularities may arise. This seems to require a notion of weak solution of the minimal surface equation. We start a discussion on this issue.
Work in collaboraton with Jens Hoppe (KTH, Sweden), Matteo Novaga (Padova, Italy), and Giandomenico Orlandi (Verona, Italy).
We consider the one-dimensional Euler-Poisson system in the repulsive regime and in Lagrangian coordinates. Assuming a sticky particle dynamics, we show that the system can be interpreted as a second-order differential inclusion on the space of optimal transport maps. We discuss a stability result for solutions of this system. Global existence then follows by approximating the initial data by finite linear combinations of Dirac measures, for which the solution can be constructed explicitly.
An estimate on the decay of Fourier space energy density of weak solutions to the Navier - Stokes equations in three space dimensions gives some restrictions on the possibilities for energy concentration. In particular it gives a lower bound on the size of the microlocal analog of the singular set, if the latter is nonempty.
A discrete analogue of the Witten Laplacian on the n‐dimensional integer lattice is considered. After rescaling of the operator and the lattice size we analyze the tunnel effect between different wells, providing sharp asymptotics of the low‐lying spectrum. Our proof, inspired by work of Helffer‐Klein‐Nier in continuous setting, is based on the construction of a discrete Witten complex and a semiclassical analysis of the corresponding discrete Witten Laplacian on 1-forms. The result can be reformulated in terms of some metastable Markov processes. These naturally arise in Statistical mechanics in the context of disordered mean field models, like the Random field Curie‐Weiss model.
We prove two facts about the binding of polarons, or its absence, which have been longstanding problems. First, the transition from many-body collapse to the existence of a thermodynamic limit for N polarons occurs precisely at $U = 2\alpha$, where $U$ is the electronic Coulomb repulsion and $\alpha$ is the polaron coupling constant. Second, if U is large enough, there is no multi-polaron binding of any kind. While these results are valid for quantized fields (the Froehlich model), in the talk we will focus on the technically easier case of classical fields (the Pekar-Tomasevich model). (The talk is based on joint work with E. H. Lieb, R. Seiringer and L. E. Thomas.)
In incompressible inviscid flow, vortex sheets are curves across which the tangential velocity is discontinuous, with zero vorticity away from the curve. They occur in many important applications, for example from the trailing edge of wings in accelerating aircraft. Most such flows have rollup of vortex sheets into algebraic spirals. Although exact solutions with logarithmic spirals have been known since Prandtl, all attempts at rigorous existence proofs of algebraic vortex spirals have been unsuccessful. We present a proof of existence in the case of large number of branches.
We investigate ground state configurations of atomic systems in two dimensions as the number of atoms tends to infinity for suitable pair interaction models. Suitably rescaled, these configurations are shown to crystallize on a triangular lattice and to converge to a macroscopic Wulff shape which is obtained from an anisotropic surface energy induced by the microscopic atomic lattice. Moreover, sharp estimates on the microscopic fluctuations about the limiting Wullf shape are obtained.
Concept of energetic solutions to rate-independent systems governed by dissipation and stored-energy functionals will be introduced and scrutinized under Gamma-convergence of those functionals. Typical situations will be illustrated on specific applications in inelastic processes in solid mechanics, as hardening plasticity limitting to perfect plasticity, adhesive contact limitting to britlle delamination, and microstructure evolution with limit passage accros various scales.
We address the nonlinear stability of a family of traveling wave solutions to the system proposed by Lane et al. (IMA J. Math. Appl. Med. Biol. 4 (1987), no. 4, pp. 309-331), to model a pair of mechano-chemical phenomena known as post-fertilization waves on eggs. The waves consist of an elastic deformation pulse on the egg’s surface, and a free calcium concentration front. These waves have been observed on the surface of some vertebrate eggs shortly after fertilization. The family of waves is indexed by a coupling parameter measuring contraction stress effects on the calcium concentration. This work establishes the spectral, linear and nonlinear orbital stability of these waves for small values of the coupling parameter. The usual methods for the spectral and evolution equations cannot be applied because of the presence of mixed partial derivatives in the elastic equation. Nonetheless, exponential decay of the directly constructed semigroup on the complement of the zero eigenspace is established. It is shown that small perturbations of the waves yield solutions to the nonlinear equations which decay exponentially to a phase-modulated traveling wave.
This is joint work with Gilberto Flores (UNAM).
Wir betrachten Flächen, die sich zu runden Punkten zusammenziehen und Netzwerke von Kurven, die gegen ein homothetisch schrumpfendes Netzwerk konvergieren.
In this talk, I will first briefly introduce the Lojasiewicz–Simon approach in the study of longtime behavior of global solutions to nonlinear evolution equations. Then as an application, we study a parabolic-hyperbolic Ginzburg–Landau–Maxwell model, which describes the behavior of a two-dimensional superconducting material. We use the extended Lojasiewicz–Simon approach to show that for any initial datum in certain phase space, the corresponding global solution converges to an equilibrium as time goes to infinity. Besides, we also provide an estimate on the convergence rate with respect to the phase space metric.In this talk, I will first briefly introduce the Lojasiewicz–Simon approach in the study of longtime behavior of global solutions to nonlinear evolution equations. Then as an application, we study a parabolic-hyperbolic Ginzburg–Landau–Maxwell model, which describes the behavior of a two-dimensional superconducting material. We use the extended Lojasiewicz–Simon approach to show that for any initial datum in certain phase space, the corresponding global solution converges to an equilibrium as time goes to infinity. Besides, we also provide an estimate on the convergence rate with respect to the phase space metric.In this talk, I will first briefly introduce the Lojasiewicz–Simon approach in the study of longtime behavior of global solutions to nonlinear evolution equations. Then as an application, we study a parabolic-hyperbolic Ginzburg–Landau–Maxwell model, which describes the behavior of a two-dimensional superconducting material. We use the extended Lojasiewicz–Simon approach to show that for any initial datum in certain phase space, the corresponding global solution converges to an equilibrium as time goes to infinity. Besides, we also provide an estimate on the convergence rate with respect to the phase space metric.In this talk, I will first briefly introduce the Lojasiewicz–Simon approach in the study of longtime behavior of global solutions to nonlinear evolution equations. Then as an application, we study a parabolic-hyperbolic Ginzburg–Landau–Maxwell model, which describes the behavior of a two-dimensional superconducting material. We use the extended Lojasiewicz–Simon approach to show that for any initial datum in certain phase space, the corresponding global solution converges to an equilibrium as time goes to infinity. Besides, we also provide an estimate on the convergence rate with respect to the phase space metric.
Wir betrachten dünnflüssige zweidimensionale Wasserwellen. Klassische Resultate zur Existenz und zur Asymptotik in Stagnationspunkten basieren auf komplexer Analysis und der Transformation in singuläre Integralgleichungen. In diesem Vortrag fokussieren wir uns auf neue geometrische Methoden, die Erweiterungen auf nicht-einfach-zusammenhängende Wasserregionen etc. zulassen.
Zentrale Resultate sind Ergebnis einer Zusammenarbeit mit Eugen Varvaruca (Imperial College London).
In this talk I will present aspects of the construction of Willmore type surfaces in asymptotically flat manifolds. The surfaces in question are critical points of the Willmore functional subject to an area constraint. The position vector of these surfaces satisfies a quasi-linear elliptic equation of fourth order. The main result ist that under suitable asymptotic conditions the asymptotic end of an asymptotically flat 3-manifold is foliated by surfaces of Willmore type that converge to Euclidean spheres as the area becomes large.
In this talk I will present Frenkel-Kontorova models which can be seen as an infinite system of coupled ODEs describing the motion of particles moving in a periodic landscape and in interaction with their nearest neighbors. The macroscopic dynamics can be seen as a conservation law (describing the conservation of particle densities).
The effective macroscopic dynamics is obtained by homogenization in the framework of viscosity solutions for Hamilton-Jacobi equations. I will present various results for overdamped or accelerated systems of particles and also for dislocations dynamics.
Wir betrachten Fragestellungen aus der Analysis, in denen Nichtlinearitäten durch unterschiedliche Mechanismen entstehen. Bei der Parametererkennung partieller Differentialgleichungen liegen diese in der Fragestellung in sich, da trotz zugrunde liegender linearer Gleichung die zu ermittelnden Größen nichtlinear von der Lösung der Gleichung abhängen.
Bei einem der klassischen Modelle der Strömungsmechanik, den Navier-Stokes-Gleichungen, entsteht die Nichtlinearität durch die Modellierung des zu beschreibenden physikalischen Phänomens. Im Vortrag soll auf Lösungsmethoden zur Behandlung dieser Fragestellungen eingegangen werden.
A fundamental feature which accounted for the success of reaction-diffusion equations is the description of spreading phenomena in unbounded domains. Estimating the asymptotic spreading speeds of solutions whose initial conditions are given is indeed one of the most important aspects from a theoretical point of view as well as in all sorts of applications in biology and ecology. In the talk, I will report on various definitions and estimates of the spreading speeds of solutions of some reaction-diffusion equations in very general domains with obstacles, or for a general class of initial conditions.
The talk is based on some joint works with H. Berestycki, N. Nadirashvili and Y. Sire.
Archimedes' principle may be used to predict when and how an object places into a bath of liquid will float. Certain inconsistencies with the predictions of the principle have been pointed out since the time of Gallileo, but have only recently been understood in something approaching a quantative manner.
I will outline previous results concerning floating objects and describe one of my own which gives some quantitative information about the shape of the liquid interface and the attitude with which a sphere may float. As an aside, I will describe the predominant reinterpretation of Archimedes' principle which likely has played a role in delaying the appearance of such results.
A simplified model for the energy of the magnetization of a thin ferromagnetic film gives rise to a version of the theory of Ginzburg-Landau vortices for sphere-valued maps. In particular we have the development of vortices as a certain parameter tends to 0. The dynamics of the magnetization is ruled by the Landau-Lifshitz-Gilbert equation, which combines characteristic properties of a nonlinear Schrödinger equation and a gradient flow. We study the motion of the vortex centres under this evolution equation.
This is a joint work with Matthias Kurzke (University of Bonn), Christof Melcher (RWTH Aachen), and Daniel Spirn (University of Minnesota).
The equilibrium shape of a crystal is determined by the minimization under a volume constraint of its free energy, consisting of an anisotropic interfacial surface energy plus a bulk potential energy. In the absence of the potential term, the equilibrium shape can be directly characterized in terms of the surface tension and turns out to be a convex set, the Wulff shape of the crystal.
Our first result is a sharp quantitative inequality implying that any shape with almost-optimal surface energy is close in the proper sense to the Wulff shape. This is a joint work with Francesco Maggi (Florence) and Aldo Pratelli (Pavia).
Under the action of a weak potential or, equivalently, if the total mass of the crystal is small enough, the surface energy of the equilibrium shape is actually close to that of the corresponding Wulff shape, and the previous result applies. However, stronger geometric properties are now expected, due to the fact that the considered shapes are minimizers. Indeed we can prove their convexity, as well as their proximity to the Wulff shape with respect to a stronger notion of distance. This is a joint work with Francesco Maggi (Florence).
Solutions of rate-independent evolution problems, as recently proposed by A. Mielke and his collaborators, can be obtained by solving a recursive minimization scheme which involves a functional governing the evolution perturbed by a suitable convex dissipation term. Rate-independence is guaranteed by the 1-homogeneity of the dissipation, which therefore has a linear growth.
The same variational scheme, with quadratic (or at least superlinear) dissipation, plays a crucial role in the variational approach to Gradient Flows.
It is then natural to investigate the relationships between these two theories, in particular when viscous approximations of rate- independent problems are considered: they are simply obtained by adding a (asymptotically small) quadratic perturbation to the dissipation term.
In this talk we address this kind of problems and we discuss some characterizations of the limit solutions obtained by general viscous approximations.
(Joint work in collaboration with A. Mielke and R. Rossi)
In this talk we shall focus on nonlinear problems that naturally arise in the study of the dynamics of populations structured by age and spatial position. We consider nonlinear diffusion and nonlinear age boundary conditions both of which depending possibly non-locally with respect to time on the population density. The abstract approach is applied to a concrete model for the swarming phenomenon of the bacterium Proteus mirabilis.
We are interested in the (asymptotic) long-time behaviour of the semigroup T_t associated with a Lévy process X_t. It turns out that Lévy processes having transition DENSITIES can be treated in a comprehensive way. In the talk we give a proof of this result and review the question when a Lévy process has a transition density.
Consider non-compact manifolds, evolving under curvature flows like Ricci flow, mean curvature flow or Gau\ss{} curvature flow. We investigate the stability of solutions to these flow equations.
The subject of this talk is a linear hyperbolic equation which arises in cosmological perturbation theory. I will describe work of Paul Allen and myself in which we determine the asymptotics of solutions of this equation. The equation is defined for positive times with a singularity at the origin. We have studied the asymptotics both in the approach to the origin and at late times. The coefficients of the equation depend on a function which represents the equation of state of a fluid. In some cases it is possible to parametrize all solutions by the coefficient functions in the asymptotic expansions in either of the asymptotic regimes.
In optical fiber communications the technique of 'dispersion management' was invented in the 90's to create stable pulses in glass fiber cables by periodically varying the dispersion along the cable. This idea turned out to be enormously fruitful in allowing for ultra high-speed data transfer through optical fibers over intercontinental distances and the dispersion management technique is now widely used commercially (as a test: google `dispersion management' and you'll get an overwelming amount of hits).
The propagation of pulses through an dispersion managed glass fiber cable is described by the Gabitov-Turitsyn equation, which is a non-local version of the non-linear Schr\"odinger equation. It has been extensively studied numerically and on the level of theoretical physics, but rigorous results are rare. We describe very recent work on the decay and regularity properties of stationary solutions of the Gabitov-Turitsyn equation.
This is joint work with Young-Ran Lee.
We present a possible approach for the computation of free energies and ensemble averages of one-dimensional coarse-grained models in materials science. The approach is based upon a thermodynamic limit process, and makes use of ergodic theorems and large deviation theory. In addition to providing a possible efficient computational strategy for ensemble averages, the approach allows for assessing the accuracy of approximations commonly used in practice.
This is joint work with X. Blanc (Univ. Paris 6), C. Le Bris (ENPC Paris) and C. Patz (WIAS Berlin).
In the talk we give a statistical motivation for variational denoising techniques in image processing. The convex techniques derived in such a way are well-known and established and rely on statistical priors and intensity errors. Correcting for sampling errors results in nonconvex variational principles which can be solved by convexification. For highdimensional data, such as medical MRI data, there is a theory based on quasi-convexification for the proposed variational problems for correcting for sampling errors. However, the existence of a quasi-convex envelope does not provide a way to numerically solve the problem.
We derive linearized theories from nonlinear elasticity theory for multiwell energies. Under natural assumptions on the nonlinear stored energy densities, the properly rescaled nonlinear energy functionals are shown to Gamma-converge to the relaxation of a corresponding linearized model. Minimizing sequences of problems with displacement boundary conditions and body forces are investigated and found to correspond to minimizing sequences of the linearized problems. As applications of our results we discuss the validity and failure of a formula that is widely used to model multiwell energies in the regime of linear elasticity. Applying our convergence results to the special case of single well densities, we also obtain a new strong convergence result for the sequence of minimizers of the nonlinear problem.
We report on a joint paper by H. Kozono, H. Sohr and R. Farwig (Acta Math. 195) dealing with the instationary Navier-Stokes equations in a general unbounded domain of $\R3$. It is known by counter-examples that the usual $L^q$-approach to the Stokes equations, well known e.g. for bounded and exterior domains, cannot be extended to general domains $\Omega\subseteq \R3$ without any modification for $q\neq 2$. However, we will show that important properties like Helmholtz decomposition, analyticity of the Stokes semigroup, and the maximal regularity estimate of the nonstationary Stokes equations remain valid for general unbounded smooth domains even for $q \neq 2$ if we replace the space $L^q$ for $2 \leq q < \infty$ by the intersection $L2 \cap L^q$ and for $1 < q < 2$ by the sum space $ L2 + L^q$. As an application we prove for general $\Omega$ the existence of a (suitable) weak solution $u$ of the Navier-Stokes equations with pressure term $\nabla p\in L_\loc^{5/4}$, conjectured by Caffarelli-Kohn-Nirenberg (1982), and satisfying both the local and strong energy inequality.
In this talk I will present a discrete model for rubber at a mesoscopic scale, based on statistical physics arguments and on a minimization principle. I will emphasize the differences with other more classical models and address the question of the derivation of a continuum limit when the mesoscopic scale is sent to zero. In particular, using the concept of stochastic lattices, I will prove the convergence towards a nonlinear elasticity model and study its mechanical properties, such as hyperelasticity, frame-invariance, isotropy and natural states. At the end, I will quickly mention related work in homogenization theory.
We consider a geometrically nonlinear model for crystal plasticity in two dimensions, with two active slip systems and rigid elasticity. We prove that the rank-one convex envelope of the condensed energy density is obtained by infinite-order laminates. We also determine the polyconvex envelope, leading to anupper and a lower bound on the quasiconvex envelope. The two bounds differ by less than 2%. We finally investigate the immpact of elastic approximations of the model.
This is joint work with Nathan Albin and Sergio Conti.
One ab initio method to simulate molecular dynamics is to use Ehrenfest dynamics, also called quantum classical molecular dynamics, where classical nuclei are coupled to quantum modeled electrons. In this talk I will derive stochastic Langevin molecular dynamics for the nuclei from Ehrenfest dynamics, at positive temperature, assuming that the molecular system is in equilibrium and that the initial data for the electrons is stochastically perturbed from the ground state.
We discuss some recent results concerning the generation of gradient flows for geodesically convex functionals in metric spaces satisfying lower bounds on their curvature. Applications are given to the Wasserstein space of probability measure and to the construction of diffusion semigroups in metric-measure spaces with Ricci curvature bounded from below.
I study the motion of a discrete interface following the minimizing movement approach, describing an effective crystalline motion in the continuum. This motion can be compared with the continuous crystalline motion described by Almgren and Taylor highlighting additional pinning, velocity quantization and non-uniqueness effects.
We study the equation $-\Delta u + \mathbf{b}\cdot\nabla u = f$ for divergence-free vector fields $\mathbf{b}$ with low regularity. Classical theory for general (not necessarily divergence-free) $\mathbf{b}$ requires at least $\mathbf{b}\in L^n$. It has been known for some time that for divergence-free coefficients, one can get results when $\mathbf{b}\in L^{n/2 +\epsilon}$. We show that for $n \geq 5$ one can even go below the exponent $n/2$, and $\mathbf{b}\in L^{(n-1)/2 +\epsilon}$ is sufficient.
In the first part of this talk I will briefly review the recent result of Tristan Rivière on the existence of a conservation law for weak solutions of the Euler-Lagrange equation of conformally invariant variational integrals in two dimensions. I will then show how we can adapt these arguments to show the existence of a conservation law for fourth order systems, including biharmonic maps into general target manifolds, in four dimensions. With the help of this conservation law I will prove the continuity of weak solutions of these systems. If time permits I will also indicate how one can use this conservation law to prove the existence of a unique weak solution of the biharmonic map flow in the energy space.
This is a joint work with Tristan Rivière (ETH Zuerich).
In this talk we consider a problem posed by J.M. Ball about the uniqueness of smooth equilibrium solutions to boundary value problems for strictly polyconvex functionals, $$ \F(u)=\int_\Omega f(\nabla u(x))\,\dd x\quad\m{and}\quad u\vert_{\de\Omega}=u_0\,, $$ where $\Omega$ is homeomorphic to a ball. We give several examples of non-uniqueness, the main of which is such a boundary value problem with at least two analytic different minimizers. All this examples are suggested by the theory of Minimal Surfaces.
Uniaxial nematic liquid crystals are often modelled using the Oseen-Frank theory, in which the mean orientation of the rod-like molecules is modelled through a unit vector field n. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which n should be equivalent to -n, that is, instead of n taking values in the unit sphere S2, it should take values in the sphere with opposite points identified, i.e. in the real projective plane RP2. The de Gennes theory respects this symmetry by working with the tensor Q=s(n\otimes n- 1/3 Id). In the case of a non-zero constant scalar order parameter s the de Gennes theory is equivalent to that of Oseen-Frank when the director field is orientable.
We report on a general study of when the director fields can be oriented, described in terms of the topology of the domain filled by the liquid crystals, the boundary data and the Sobolev space to which Q belongs.
We also analyze the circumstances in which the non-orientable configurations are energetically favoured over the orientable ones.
This is joint work with John Ball.
We will discuss new results about approximation of Sobolev mappings between manifolds and metric spaces. We will also discuss degree theory for Sobolev mappings in the Orlicz-Sobolev space slightly below $W^{1,n}$.
For a map between Riemannian manifolds, the tension field is the L2-gradient of the Dirichlet functional. Consider the second order functional given by the L2-norm of the tension field. Its critical points are called (intrinsic) biharmonic maps.
When we want to apply methods from the calculus of variations, such as the direct method, to this problem, we face a number of difficulties. Most importantly, the functional is not coercive on the natural Sobolev spaces. We discuss an approach to the problem via a relaxation of the functional, using tools from geometric measure theory.
We study the decay and existence of solutions to some equations modeling polymeric flow. We consider the case when the drag term is corotational. We analyse the decay when when the space of elongations is bounded, and the spatial domain of the polymer is either a bounded domain $\Omega \subset \rn,\;\;n=2,3$ or the domain is the whole space $\rn,\;\;n=2,3$. The decay is first established for the probability density $\psi$ and then this decay is used to obtain decay of the velocity $u$. Consideration also is given to solutions where the probability density is radial in the admissible elongation vectors $q$. In this case the velocity $u$, will become a solution to Navier-Stokes equation, and thus decay follows from known results for the Navier-Stokes equations.
Some questions in relation to Poincaré type inequalities, and fluid equations in general, will be discussed
The spreading of thin liquid films is described by the thin-film equation. It is a degenerate parabolic equation of fourth order, describing a free boundary problem.
Existence, but not uniqueness, for weak solutions have been proved. We show existence and uniqueness for classical solutions. The solution space is a weighted Sobolev space.
The Modica-Mortola functional of phase transitions is well-known to Gamma converge to the hypersurface area functional as the small parameter tends to zero. It is now known that the control of mean curvature-like quantity is sufficient for the convergence, and more subtle questions can be asked about its property. I review some known results along with open problems and discuss related problems such as Allen-Cahn equation with or without coupled terms, Cahn-Hilliard equation and a functional motivated by the large deviation theory. Some of the questions are of technical and others of intrinsic geometric nature in the setting of geometric measure theory.
It will be explained how diffusion equations enter the field of image denoising and enhancing and discuss, in particular, the famous Perona - Malik problem. Then we motivate and propose time-regularized well-posed versions of the Perona-Malik equations and give numerical evidence for their superiority to the widely used space regularizations.
We study the reklationship between Sogge's $L^p$ spectral estimates for the Laplace operator and Strichartz type estimates for the wave operator. As a consequence of this analysis, we obtain that in 3-d domains, the quintic (critical) defocussing wave equation with Dirichlet boundary conditions is globally well posed in the energy space. This extend previous results of Grillakis and Shatah-Struwe on $R^d$. (joint with G. Lebeau and F. Planchon)
Via gauge theory, we give a new proof of partial regularity for stationary harmonic maps into arbitrary targets. This proof avoids the use of adapted frames and permits to consider targets of "minimal" $C2$ regularity. The proof we present extends to a large class of elliptic systems of quadratic growth.
We study the Gamma-convergence of functionals arising in the Van der Waals-Cahn-Hilliard theory. The corresponding limit functional is given as the sum of the area and the Willmore functional. The problem under investigation was proposed as modification of a conjecture of De Giorgi.
Dislocations are topological defects in crystal that are considered responsible for plastic deformations. We consider a 2D model for edge dislocations, where the deformation has a singularity on points that represent dislocations, while the crystal behaves elastically far from the core. This model is very close to the 2D Ginburg-Landau model for the study of vortices in superconductors. We study, in a dilute regime, the limit as the number of points (dislocations) tends to infinity and we obtain as limit problem an elasto-plastic model, given by the elastic energy and a term depending on the Curl of the plastic deformation (the dislocations density).
The 3D-Navier-Stokes system for a steady compressible isentropic fluid, with a pressure law $p=\rho^\gamma$ in a bounded domain is considered. We extend the existence theory to the range $\gamma\geq 4/3$ in isomass case and to the range $\gamma>1$ in the isoenergy case.
I shall discuss the influence of the geometry of the medium onto the wellposedness theory of the Cauchy problem for nonlinear Schroedinger equations by surveying a series of results on spheres obtained in collaboration with Nicolas Burq (Orsay) and Nikolay Tzvetkov (Lille).
We study the steady solutions of Euler-Poisson equations in bounded domains with prescribed angular velocity. This models a rotating Newtonian star consisting of a compressible perfect fluid with given equation of state $P=e^S\rho^{\gamma}$. When the domain is a ball and the angular velocity is constant, we obtain both existence and non-existence theorems, depending on the adiabatic gas constant $\gamma$. In addition we obtain some interesting properties of the solutions; e.g., monotonicity of the radius of the star with both angular velocity and central density. We also prove that the radius of a rotating spherically symmetric star, with given constant angular velocity and constant entropy, is uniformly bounded independent of the central density . This is physically striking and in sharp contrast to the case of the nonrotating star. For general domains and variable angular velocities, both an existence result for the isentropic equations of state and non-existence result for the non-isentropic equation of state are also obtained.
I will discuss a method to represent curves evolving under curve shortening flow as nodal sets of the limit of solutions to the parabolic Ginzburg-Landau equation. For any given compact curve $\Gamma$ that satisfies the equation \be \frac{\partial \Gamma}{\partial t}=k_\Gamma\hat{n},\label{csf1}\ee I construct a family of solutions to the parabolic Ginzburg-Landau: \be \frac{\partial u_\epsilon}{\partial t}-\Delta u_\epsilon +\frac{(\nabla_u W)(u_\epsilon)}{2\epsilon ^2}=0 \label{gl11}\ee such that for $\bar{t}
We review the basic properties of the degenerate and singular evolution equation \[ u_t=\left(D2 u \frac{Du}{\abs{Du}}\right)\cdot\frac{Du}{\abs{Du}}, \] which is a parabolic version of the increasingly popular infinity Laplace equation. We discuss how to obtain existence and uniqueness for both Dirichlet and Cauchy problems, establish interior and boundary Lipschitz estimates and a Harnack inequality, and also provide some interesting explicit solutions. This is a joint work with Bernd Kawohl, Cologne.
We study the evolution of passive scalar fields maintained by steady but spatially inhomogeneous sources and sinks, and stirred by an incompressible flow. The effectiveness of a flow field to enhance mixing over molecular diffusion is measured by the suppression of the space-time averaged scalar variance, the gradient variance (stressing small scales), and the inverse gradient variance (focusing on large scale fluctuations). Ratios of these variances without stirring to the corresponding variances with stirring provide non-dimensional measures of the "mixing efficiency" of the flow on different scales. In this work we derive rigorous estimates on these multi-scale mixing efficiencies for a variety of source distributions with general stirring flows including statistically homogeneous and isotropic velocity fields, and compare them with direct numerical simulations and exact calculations for sample problems.
This is joint work with Matt Finn and Jean-Luc Thiffeault from Imperial College, and Tiffany Shaw from the University of Toronto.
The class of mappings of finite distortion may serve as the space of possible deformations in continuum mechanics. In this talk, the question of invertibility of mappings of finite distortion is discussed. The main result says that the inversion of a homeomorfism of finite distortion with gradient in the Lorentz space L(n-1,1) is a mapping of finite distortion.
In 1968 V.E. Zakharov derived the Nonlinear Schrdinger equation for the 2D water wave problem in case of no surface tension, i.e. for the evolution of gravity driven surface water waves, in order to describe slow temporal and spatial modulations of a spatially and temporarily oscillating wave packet. Since this time the question remained open if solutions of the 2D water wave problem behave as predicted by the Nonlinear Schrdinger equation. Here we answer this question positively.
This is joint research with C.E. Wayne (Boston)
We show that every $L^r$-vector field on $\Omega$ can be uniquely decomposed into two spaces with scalar and vector potentials and the harmonic vector space via $rot$ and $div$, where $\Omega$ is a bounded domain in $R3$. As an application, the generalized Biot-Savard law for the invisid incompressible fluids in $\Omega$ is obtained. Furthermore, we prove a blow-up criterion such as Beale-Kato-Majda type via vorticity in $bmo$ on the classical solution of the Navier-Stokes equations in $\Omega$.
Hyperbolic systems of conservation laws are naturally linked to transport equations with variable nonsmooth coefficients. In particular, relying on recent progress on transport equations, it is possible to give an higher dimensional existence theory which handles singularity formation for some very special systems. In this talk we address three questions on transport equations which naturally arise in trying to extend this existence theory to more general systems.
The study of Sobolev type isometric immersions between Riemannian manifolds is motivated from different perspectives. The talk will put forward various problems of geometric or analytical nature arising in this context. Some partial regularity and rigidity results for certain classes of Sobolev isometric immersions will be discussed.
The symmetric minimal surface equation (SME) on a domain $\Omega\subset{}R^{n}$ is the equation ${\cal{}M}u={m-1\over{}u\sqrt{1+|Du|^{2}}}$, where $m$ is an integer $\ge 2$ and ${\cal{}M}(u)$ is the mean curvature operator. Geometrically this equation expresses the fact that the symmetric graph $S(u)=\{(x,\xi)\in \Omega\times{}R^{m}:|\xi|=u(x)\}$ is a minimal (i.e.\ zero mean curvature) hypersurface in $R^{n+m}$. For $n\ge 2$ the SME admits singular solutions (solutions which vanish at some points but which are nevertheless locally the uniform limit of positive smooth solutions), and such singular solutions have symmetric graphs which are singular minimal hypersurfaces. The talk will develop the theory of such singular solutions; both regularity theory (including gradient bounds) and existence theory will be discussed.
We consider the first variational problem rigorously solved by the direct method, in Tonelli's pioneering work on convex and super linear energies in the one dimensional situation. Beside the existence, he also proved the first partial regularity result for the minimisers. Optimality of his work, i.e. the existence of a large class of singular sets for minimisers was established over a longer period - culminating in the work by John M. Ball and others. There also the notion of universal singular sets (i.e. points singular for some boundary data) was introduced; later studied by M.Sychev. A quite sharp characterisation of these sets will be discussed here.
In this talk I will consider dislocations which are line defects in crystals. When a stress is applied on the crystal these lines can move. I will introduce a mathematical modelling of dislocation dynamics as non-local eikonal equations. These equations are studied in the framework of viscosity solutions. This model arises new questions and I will give some recent results.
The Gross-Pitaevsky equations serve as an excellent model for superfluids. One important feature of superfluids is the formation of very tight vortices that rotate with the liquid helium. In the theoretical model, a very tight vortex corresponds to a very large Ginzburg-Landau constant. In the limit of large Ginzburg-Landau constant, it has been previously shown that these vortices move according to the same classical ODE as point vortices do in two-dimensional incompressible, inviscid fluids. I will present joint work with R. Jerrard concerning the proof of this motion law for large, but fixed, Ginzburg-Landau constant. The primary tools will be sharp and near sharp estimates of the Jacobian.
In analogy with fluid mechanics, to the superconducting current in the model of Ginzburg and Landau can be associated a vorticity, which in the extreme type II limit and under natural hypothesis converges to a measure. Over the years, in joint work with Sylvia Serfaty, we have studied these measures. We describe here some old and recent results, as well as open problems.
(joint work with Dirk Hundertmark, Rowan Killip, Shu Nakamura, Peter Stollmann) We derive three results belonging to the theory of Schroedinger operators. First we consider a pair of Schoedinger operators $H_1, H_2$ differing by a compactly supported potential $V$. We show that the singular values $a_n$ of the difference of exponentials $e^{-t H_2} -e^{-t H_2}$ decay almost exponentially as $n$ tends to infinity. Thereafter this result is used to derive an upper bound on the spectral shift function $\xi (E, H_2, H_1)$. This function captures how much of the spectral density is shifted across the energy $E$ by the perturbation $V$. Our upper bound is close to a lower bound which can be established by an example. Finally we apply the spectral shift bound to prove a Wegner estimate for certain random Schroedinger operators called alloy-type models. It implies that the integrated density of states (=spectral density function) is Hoelder continuous. Our continuity requirements on the randomness entering the operator are weaker than the ones needed for earlier proofs of Wegner estimates.
As an example one can take the Laplace equation plus the square of the unknown, $-\Delta u + u2=f$ in an open set $\Omega$ in $R^n$, considered with a Dirichl\'et condition $u=g$ on the boundary. The purpose of the talk is to explain how one can obtain parametrices $P_N$ of this non-linear problem. The resulting parametrix formula $u=P_N(Rf+Kg)+ (RL)^Nu$ expresses a given solution $u$ via terms depending on the data $(f,g)$ \[$R$, $K$ are solution operators of the corresponding linear problem\] plus a remainder in $C^k$ for arbitrarily large $k$. The formula implies that solutions belong to the same spaces as in the linear case, under some mild assumptions allowing non-classical cases in which the solution `ends up' in a space on which the non-linear term $u2$ is ill-defined. The parametrix construction uses pseudo- and paradifferential techniques, and it extends to general semi-linear elliptic systems with non-linear terms of product type.
We discuss unstable free boundary problems arising in SHS/gasless combustion and smoldering combustion. We focus on a result obtained in collaboration with Régis Monneau (CERMICS, Paris) concerning the stationary equation \[ \Delta u = -\chi_{\{ u>0\}}\] arising as first order approximation in solid combustion: some partial regularity holds for maximal solutions of this equation. In particular, in two space dimensions the interface consists locally of a finite number of $C1$-arcs meeting in quadruple junctions.
The motivation for this work are "period-doubling" bifurcations of spiral waves in excitable media. The bifurcating spirals exhibit line defects across which the temporal phase of the period-doubling mode jumps. If there are several line defects present, then they begin to interact and, in fact, attract each other. Interacting coherent structures have been observed in many other situations as well. After giving an overview of the different types of coherent structures, I will describe an attempt to explain (some of) their interaction properties phenomena by spectral analyses.
In the joint work with G. Leoni and M. Morini we present a general theory to study optimal regularity for a large class of nonlinear elliptic systems satisfying general boundary conditions and in the presence of a geometric transmission condition on the free-boundary. As an application we give a full positive answer to a conjecture of De Giorgi on the analyticity of local minimizers of the Mumford-Shah functional.
The pressure term has always created difficulties in treating the Navier-Stokes equations of incompressible flow, reflected in the lack of a useful evolution equation or boundary conditions to determine it. In joint work with Jian-Guo Liu and Jie Liu, we show that in bounded domains with no-slip boundary conditions, the Navier-Stokes pressure can be determined in a such way that it is strictly dominated by viscosity. As a consequence, in a general domain with no-slip boundary conditions, we can treat the Navier-Stokes equations as a perturbed vector diffusion equation instead of as a perturbed Stokes system. We illustrate the advantages of this view by providing simple proofs of (i) the stability of a difference scheme that is implicit only in viscosity and explicit in both pressure and convection terms, requiring no solutions of stationary Stokes systems or inf-sup conditions, and (ii) existence and uniqueness of strong solutions based on the difference scheme.
In the framework of the "principle of the fermionic projector" the following variational principle was proposed, \[ \sum_{x,y \in M} {\mathcal{L}}[P(x,y)\: P(y,x)] \;\rightarrow\; \min\:, \] where the "Langrangian" ${\mathcal{L}}$ is given by \[ {\mathcal{L}}[A] \;=\; |A|^2 - \mu |A^2| \:. \] Here~$P$ is the fermionic projector, $M$ denotes the points of discrete space-time, $\mu$ is a Lagrangian multiplier and~$|.|$ is the so-called spectral weight. The purpose of the talk is to motivate this variational principle and to explain some analytical aspects. In particular, I want to work out in which sense the vacuum is a stable minimizer of this variational principle. The talk is intended for analysts, no knowledge about mathematical physics is required.
We are concerned with the problem of defining a reasonable notion of global solution to the partial differential equation obtained as the $L2$-gradient flow of an integral functional with an integrand $\phi$ which is nonconvex in the gradient $u_x$. Typical examples of integrands are $\phi(u_x) = \log(1+u_x2)$ and $\phi(u_x) = (1-u_x2)2$, which give raise to backward-forward parabolic equations. We discuss some results related to the approximation of such equations; we construct also a solution for a particularly simple nonconvex integrand. These results are part of a series of papers (mainly in progress) in collaboration with G. Fusco (Univ. l'Aquila), N. Guglielmi (Univ. l'Aquila), M. Novaga (Univ. Pisa), E. Paolini (Univ. Firenze).
Among all surfaces of convex bodies the sphere is shown to be unique with respect to the property that, in the absence of exterior fields, electric charges distribute homogeneously on the surface. The proof involves maximum principles, the isoperimetric inequality and convex analysis.
We present a new analytical tool to quantitatively estimate the waiting time for free boundary problems associated with degenerate parabolic equations and systems. Our approach is multi-dimensional, it applies to a large class of equations, including thin-film equations, (doubly) degenerate equations of second and of higher order and also systems of semiconductor equations. For these equations, we obtain lower bounds on waiting time which we expect to be optimal in terms of scaling. This assertion is true for the porous medium equation which seems to be the only PDE for which quantitative estimates of the waiting time have been established so far.
This is joint work with L. Giacomelli, Roma.
This lecture contains variations on the p-Laplace operator as p goes to 1 or to infinity. It turns out that in the limit one arrives at interesting geometric problems. I shall report on geometric results for instance for principal eigenfunctions, for a class of Bernoulli problems and for some overdetermined boundary value problem.
Let $f$ be a function from ${\Bbb R}$ to ${\Bbb R}$ with $f(0)=0$, continuous, increasing, and differentiable at zero. Let ${\Bbb R}^d$ with $d\ge 2$. For every open set $U\subset X$, we set: ${\cal H}_f(U)=\{u\in {\cal C} (U):\Delta u=f(u)\}$ in the distributional sense. We define a function in ${\cal H}_f(U)$ tending to infinity at the regular boundary of $U$ to be a regular Evans function associated with $f$, $U$, and $\Delta $. We'll say that $f$ has the KO (Keller-Osserman) property if there exists some natural number $d\ge 2$ such that a regular Evans function associated with $f$, $B$, and $\Delta $ for every ball $B\subset \Bbb R^d$. We'll give an explicit characterisation of that property and examine the relationship between the KO condition, the Harnack principle, and the Brelot convergence property. We prove that, in the nonlinear case --- and in contrast to the linear case, we do not have the equivalence between the latter two properties. We continue the investigation of regular Evans functions in the case of uniformly elliptic or uniformly parabolic operators where we replace the function f by a function $\psi $ from ${\Bbb R}^d \times {\Bbb R}$ to ${\Bbb R}$ which, in contrast to many other authors, we do not suppose to be convex or locally Lipschitz. We then show that the previous investigations lead in a simple and natural way to results for the following generalized Ginzburg-Landau equation: \begin{equation} Lu-u(|u|^{2\alpha}-1)=0 \,\, in the distributional sense on \,\, {\Bbb R}^d \end{equation} where $\alpha$ is a positive real number and L is a strongly elliptic operator with bounded, uniformly Hölder-continuous coefficients, admitting an adjoint $L^{\ast }$ in the distributional sense. We shall obtain Hervé and Hervé-Harnack inequalities and discuss the solvability of the Dirichlet problem for real and complex valued solutions for the Ginzburg-Landau equation.
Self-similarity is a frequently used argument in mathematics and physics. We will focus on the scaling of a self-similar Dirichlet energy (the quadratic form of a Laplacian) on a self-similar graph. This imposes a self-similar geometry on the graph which allows a nontrivial heat conduction. The corresponding existence and uniqueness results can be phrased in the framework of nonlinear Perron-Frobenius Theory. Applied to the graphical skeleton of self-similar (finitely ramified) fractal set this technique enables us to couple certain fractals with different scaling laws. The resulting range of admissible scalings on the interface models different types of transition. Its analytic meaning is a scale of Besov spaces appearing as the trace (on the interface) of the set of functions of finite energy.
The stationary front in the title represents a planar interface which separates the two stable phases of a fluid. Due to stochastic forces, its position fluctuates and we would like to determine the most probable way in which a macroscopic displacement $R$ may occur in a given macroscopic time interval $T$. Supposing planar symmetry, the problem becomes one dimensional and it is modelled by introducing a non local cost functional which must then be minimized over orbits which exhibit the desired displacement in the given time.It is found that in a "sharp interface limit" the optimal behavior for $R$ small enough, is when the front moves with constant velocity $V=R/T$, the corresponding cost being $c V^2T$, $c$ a positive constant. However when $R$ increases past a critical value, the cost becomes smaller than $c V^2T$. The effect is caused by nucleations ahead of the moving front; there are critical values $R_n$, $n\ge 1$, so that, if $R\in (R_n,R_{n+1})$ then there are $n$ nucleations.
We present a regularity result for kinetic averaging of solutions of kinetic equations, with no a priori assumption on the solution itself, but only assumptions on the data of the problem (source term, initial condition). First, we will recall the history of "kinetic averaging lemmas" in the litterature and show the originality of our result. Then, we will explain our main motivation for such a result : how kinetic interpretation can be useful to understand hyperbolic problems such as conservation laws and micromagnetism. Finally, we will give ideas of proof.
I will discuss results obtained jointly with G. Bellettini and E. Presutti. We study "tunnelling" in a one-dimensional, non local evolution equation, by assigning a penalty functional to orbits which deviate from solutions of the evolution equation. We discuss the variational problem of computing the minimal penalty for orbits which connect two stable, stationary solutions.
We consider the initial-boundary value problem in a convex domain for the Vlasov-Poisson system. Boundary effects play an important role in such physical problems that are modeled by the Vlasov-Poisson system. We establish the global existence of classical solutions with regular initial boundary data under the absorbing boundary condition. We also prove that regular symmetric initial data lead to unique classical solutions for all time in the specular reflection case.
We study the continuation of solutions of superlinear indefinite parabolic problems after the blow-up time. The nonlinearity is of the form $a(x)u^p$, where $p>1$ is subcritical and $a$ changes sign.
This talk concerns numerical discretizations of surface diffusion which is concerned with the evolution of a surface with normal velocity being the surface Laplacian of the mean curvature.
One of the key issues related to superfluidity is the existence of quantized vortices. We present very recent experiments on Bose-Einstein condensates exhibiting vortices, which consist in rotating the trap holding the atoms. We investigate the behavior of the wave function which minimizes the Gross Pitaevskii energy. This energy takes into account the special shape of the trapping potential. In a regime with a small parameter, we give a simplified expression of the energy which only depends on the number and shape of vortex lines. This allows us to study in detail the structure of the lines which have either a $U$ or $S$ shape and compare with experiments. We also present results where the type of trapping potential can be at the origin of multiply quantized vortices. Finally, we will mention another type of experiments where the condensate flows around an obstacle and we are interested in existence of vortex free solutions.
The Euler-Poincare equations were born in 1901 when Poincare made a sweeping generalization of the classical Euler equations for the rigid body and ideal fluids. He did this by formulating the equations on a general Lie algebra, with the rigid body being associated with the rotation Lie algebra and fluids with the Lie algebra of divergence free vector fields. Since then, this setting has been used for many other situations, such as the KdV equation, shallow water waves, averaged fluid equations, and the template matching equations of computer vision to name just a few. This talk will give an overview of this general approach and then will focus on the specifics for the case of the algebra of all vector fields. Special singular solutions will be described which generalize the peakon (soliton) solutions of the shallow water equations from one to higher dimensions; it will be shown that momentum maps (in the sense of Noether's theorem from mechanics) play an important role in these singular solutions. (Joint work with Darryl Holm)
A method for constructing non C^1 minimizers of strictly polyconvex integral functionals will be given. The method relies on a simple geometric characterization of the zero-sets of strictly polyconvex functions.
Understanding self-contact in elastic rod theory is difficult in large part due to the potentially complex geometry. We study a simplified problem in which the freedom of movement of the rod is restricted: the centerline of the rod is constrained to lie on a cylindrical surface. Under common loading conditions the rod will try to penetrate itself; with an appropriate non-interpenetration condition this yields a (non-local) contact problem.
We investigate geometrically exact generalized continua of Cosserat micropolar type. The basic difference of the Cosserat model to classical models for solids is the appearance of a field of independent rotations. We introduce the Cosserat model in variational form as a minimization problem. The model naturally incorporates an internal length scale which is characteristic of the material, e.g. the grain size and which is supposed to be a localization limiter. This length scale also introduces size effects to the extent that small samples of a material behave comparatively stiffer than large samples.It is motivated that the traditional Cosserat couple modulus $\mu_c$ appearing in the model can and should be set to zero for macroscopic specimens liable to fracture in shear, still leading to a complete consistent Cosserat theory with independent rotations in the geometrically exact finite case in contrast to the infinitesimal, linearized Cosserat model. Depending on material constants different mathematical existence theorems in Sobolev-spaces are given for the resulting nonlinear boundary value problems in the elastic case. Partial focus is set to the possible regularization properties of micropolar Cosserat models compared to classical continuum models in the macroscopic case of materials failing in shear.The mathematical analysis heavily uses an extended Korn's first inequality (Neff, Proc.Roy.Soc.Edinb.A, 2002) discovered recently. The methods of choice are the direct methods of the calculus of variations. An analytical example shows the beneficial effect of independent rotations and the possibility to describe sharp interfaces.
The study of dislocations in cristalline solids involves phenomena taking place at different interacting scales: from the nanoscale (a few atomic spacings, the width of dislocation cores) to the macroscale, where the collective behavior of dislocation densities controls properties such as the strength of the material. We address here the main issue at the atomic scale: the understanding of the structure and mobility of isolated dislocations.
We present a discrete model for the dynamics of dislocations in cubic metals. Numerical solutions of the model in 3D suggest that dislocations can be identified with discrete nonlinear waves. In simplified 1D and 2D geometries, we are able to obtain information about the depinning thresholds (dynamic and static Peierls stress) and the speed of the defects thanks to the analysis of traveling wave solutions of the discrete models.
Let $\dist_K$ be the distance from a compact set $K\subset\matrices$ in the space of $m\times n$ matrices. This note determines the set $M_p\subset\matrices$ of zeroes of the polyconvex hull of $\dist_K^p$ where $1\leq pn/2$ and/or $s=1.$ In the remaining cases only bounds are obtained. A surprising consequence is that the quasiconvex hull of $\dist_{SO(n)}^p$ is not polyconvex if $1\leq p
We consider a dynamical system describing the diffusive motion of a particle in a double well potential with a periodic perturbation of very small frequency, and an additive stochastic perturbation of small amplitude. It is in stochastic resonance if the solution trajectories amplify the small periodic perturbation in a 'best possible way'. Systems of this type first appeared in simple energy balance models designed for a qualitative explanation of global glacial cycles. Large deviations theory provides a lower bound for the proportion of the amplitude and the logarithm of the period above which quasi-deterministic periodic behavior can be observed. To obtain optimality, one has to measure periodicity with a measure of quality of tuning. Notions of quality of tuning widely used in physics such as the spectral power amplification or the signal-to-noise ratio depend on the spectral properties of the averaged trajectories of the diffusion. These notions pose serious mathematical problems if the underlying system is reduced to simpler Markov chain models on the finite state space composed of the meta-stable states of the potential landscape in the limit of small noise. As a way out of this dilemma we propose to measure the quality of periodic tuning by the probability that transitions between the domains of attraction of the potential wells happen during a parametrized time window maximized in the window parameter. This notion can be investigated by means of uniform large deviations estimates and turns out to be robust for the passage to dimension reduced Markov chains.
Non-constant analytic functions map open sets to open sets and the preimage of a point is a discrete set of points. A natural generalization of the concept is that of a mapping of bounded distortion in the n-dimensional euclidean space: a continuous mapping whose first order distributional derivatives are n-integrable and so that the n-th power of the norm of the differential matrix Df(x) is almost everywhere controlled by a constant K multiple of the Jacobian determinant. By a result of Reshetnyak, the disceteness and openness holds for mappings of bounded distortion. We discuss (optimal) extensions of this result to the setting where K is allowed to depend also on the variable x. A mapping constructed by Ball appears to give the critical regularity of K.
Let $f$ be a bi-Lipschitz mapping of the Euclidean ball $B_{\mathbb{R}^n}$ into $\ell_2$ with both Lipschitz constants close to one. We investigate the shape of $f(B_{\mathbb{R}^n})$. We give examples of such a mapping $f$, which has the Lipschitz constants arbitrarily close to one and at the same time has in the supremum norm the distance at least one from every isometry of $\mathbb{R}^n$.
Separation of scales plays a fundamental role in the understanding of the dynamical behavior of complex systems in physics and other natural sciences. A prominent example is the Born-Oppenheimer approximation in molecular dynamics. In my talk I will present a general scheme for deriving effective equations of motion for the slow degrees of freedom in a quantum mechanical system with two time-scales. As an application I consider higher order corrections to the conventional Born-Oppenheimer approximation in molecular dynamics.
Transversality of stable and unstable manifolds in the infinite diemsional context will be discussed. A proof of its genericity for reaction diffusion and damped hyperbolic equations will be outlined.
We consider the thin film approximation of a 1-d scalar conservation law with strictly convex flux. We prove that the sequence of approximate solutions converges to the entropy solution. (Joint work with Felix Otto)
We shall consider the kinetic equation describing the Compton scattering interaction between electrons and photons. We will see that the solutions may concentrate at the origin and form a Dirac mass assymptotically as $t$ tends to infinity. This is a sort of Bose condensation in infinite time. We will describe precisely the formation of the Dirac mass and show that it is of self similar form and strongly depends on the behavior of the initial data near the origin.
We propose a new approximation scheme for the Stefan problem with Gibbs-Thomson law. Within a time-discretization given by Luckhaus we choose approximate phase functions by a local instead of a global minimization. Driving the approximations to a limit we have to deal with a possible loss of surface area of the phase interfaces. The idea is to investigate the convergence of the surface measures. We formulate a generalization of the Gibbs-Thomson law and prove long-time existence of solutions using a convergence result due to Schätzle.
The time-dependent Hartree-Fock (TDHF) equations describe the evolution of large systems of fermions (e.g. electrons) in the mean-field limit. They play an important role for instance in Chemistry. In this talk, we explain how this system can be rigorously derived from the linear, N-body Schroedinger equation with pairwise interaction under a scaling assumption known as "weak coupling".
(This result has been obtained in collaboration with C. Bardos, A. Gottlieb and N. Mauser).
We consider the Yamabe or scalar curvature flow on general compact closed manifolds. By showing convergence of the scalar curvature to its average value in all L^p norms for t → ∞, we deduce via a concentration-compactness arguement that the metrics either converge to a smooth Yamabe metric, or else concentrate in finitely many bubbles. In the presence of at most one bubble we identify a Kazdan-Warner type transversality condition that rules out concentration and therefore implies convergene of the flow. The condition is very natural and easily verified when the manifold is conformal to the standard sphere. Using the positive mass Theorem we proof that the criterion also holds on general manifolds of dimensions 3 ≤ n ≤ 5 and in the local conformally flat case.
This is joint work with M. Struwe, Zürich
Variational integrals modeling solid-to-solid phase transformations often fail to be weakly lower semicontinuous because the energy densities f are not quasiconvex in the sense of Morrey. In this talk we analyse properties of minimizing Young measures generated by minimizing sequences for these variational integrals. We prove that the moments of order q>p exist if the integrand is sufficiently close to the p-Dirichlet energy at infinity. A counterexample related to the one-well problem in two dimensions shows that one cannot expect in general L∞ estimates, i.e., that the support of the minimizing Young measure is uniformly bounded.
This is joint work with J. Kristensen and K. Zhang.
The Kuramoto--Shivashinsky (KS) equation
u_t + (1/2 u^2)_x + u_{xx} + u_{xxxx} = 0
is a Burgers equation with a destabilizing second and stabilizing fourth order term. One interested in the long--time behavior of periodic mean--zero solutions with period L>>1. Solutions in numerical simulations show a chaotic behavior and are of O(1) in L. Rigorous results are not abundant. Together with Lorenzo Giacomelli, we slightly improve the bound spatial average of |u| \le O(L^{11/10}) by Eckmann et. al. to spatial average of |u| \le o(L) (little o). This is still far from optimal, but at least shows that KS behaves differently than ``Burgers--Shivashinski''
u_t + (1/2 u^2)_x - u - u_{xx} = 0,
which looks similar to KS, but admits periodic solution u of order O(L).
We discuss quasistatic initial-boundary value problem with internal variables which model the deformation behavior of viscoplastic bodies at small strains. In the first part of the talk we introduce the class of constitutive equations of monotone type, discuss the relevance of this class to engineering applications and study the existence theory. In the second part of the talk these results are used to justify the formal homogenization of such initial-boundary value problems with periodic microstructure, called the microscopic problems. To this end it is first shown that the formally derived homogenized initial-boundary value problem has a solution. From this solution an asymptotic solution of the microscopic problem is constructed, and it is shown that the difference of the exact solution and the asymptotic solution converges to zero if the lengthscale of the microstructure tends to zero. This convergence proof is based on the notion of operator difference introduced by Vladimirov.
We consider an elliptic regularization of the classical Perona-Malik functional (in 1-D) obtained by adding a small quadratic term depending on the second derivative. After a suitable rescaling we characterize the Gamma-limit for vanishing value of the singular parameter. We then introduce a "slow time" and study the gradient dynamic determined by the Gamma-limit functional. We also discuss some numerical experiments and some relationship between the gradient dynamic corresponding to the regularized functional and the one corresponding to the Gamma-limit.
We consider a binary mixture of fluids interacting with a repulsive long range force between different species. Masses, total momentum and energy are conserved. The time evolution is described by a couple of Boltzmann equations for the phase space densities of the two species. Under appropriate space-time scalings hydrodynamic equations of Euler and Navier-Stokes type are obtained, with additional self-consistent forces. The equilibria are described on all the scales as minimizers of an appropriate free energy. The minimizers, corresponding to segregated phases at low temperature, are determined, in the kinetic scale, under the assumption of monotonicity of the potential, via rearrangement inequalities and a local analysis. A formal asymptotic expansion for the evolution of a sharp interface is obtained.
The motion of two non-mixing fluids in a porous material is commonly described by the two-phase-flow or Leverett equations. Capillarity induces a pressure difference between the two fluids, the capillary pressure. We propose and analyze a model for the motion of fronts in a porous material, aiming at the justification of this pressure jump. We work on a meso-scale in which the single front is still resolved, and homogenize the effects of the single pores. Stationary methods and the methods of periodic homogenization both turn out not to be suited for this problem. Using the language of micro-local-patterns we can formulate a conditional result for upscaled equations; in the stochastic case the upscaled system is satisfied almost surely.
Deterministic bifurcation theory deals with qualitative changes in parametrized families of vector fields or dynamical systems. "Qualitative change" has been successfully formalized by means of topological equivalence and structural stability. We review the generic elementary bifurcation scenarios.
Stochastic bifurcation theory should deal with qualitative changes in parametrized families of random dynamical systems (systems perturbed by noise), and should reduce to the deterministic theory if noise is absent. While the concept of a random dynamical system can be unanimously formalized, this is not the case with "qualitative change". I offer a "phenomenological approach" introduced by physicists already in the eighties, and a "dynamical approach" based on Lyapunov exponents, invariant measures and random attractors.
We then try to find the stochastic versions of the elementary bifurcation scenarios, in particular of the Hopf bifurcation, mainly by way of prototypical examples like the Duffing-van der Pol oscillator with multiplicative noise and the Duffing (Kramers) oscillator with additive noise.
We study the homogenization of functionals of the form These functionals are related to a model of dielectric breackdown and can be used as a variational approximation of power-law energies. We compute optimal bounds in the case of the mixtures of two materials, which exhibit a different structure from the integral case. We prove a homogenization theorem in a general context and compare it with the corresponding result for the integral case.
We study the wellposedness of an eikonal equation with a discontinuous right hand side. Under suitable assumptions on the data we obtain a unique solution, which satisfies the equation in the (suitably extended) viscosity sense. We also analyze a finite difference scheme for approximating this solution.
In this talk, we shall discuss a class of singular sets associated with differential equations. Such sets are simply the critical nodal sets for elliptic equations of the second order. They form the singular parts of the nodal sets. We shall show that these sets are rectifiable of codimension 2 under very few assumptions on the coefficients of equations. We shall also discuss how to obtain measure estimates of these sets. Methods are from geometric measure theory and several complex variables.
In the talk we explain an operator-theoretic approach to the Korteweg-de Vries equation. Using the theory of determinants on quasi-Banach ideals as fundamental tool, we derive a solution formula, which contains an operator-valued parameter.
It can be shown that all solutions covered by the Inverse Scattering Method (and more) can be realized in this frame.
As an application we discuss negatons, a solution class with the following properties: They consist of solitons which are organized in groups, and solitons in the same group are weakly coupled. In contrast groups as a whole show a particle-like behaviour. Within our formalism negatons corrsepond to the case of finite-dimensional parameters.
Our main result is the complete characterization of negatons in terms of their long-time behaviour.
A nonlocal model of non-isothermal phase separation in binary alloys is presented. The model is deduced from a free energy with a nonconvex part taking nonlocal particle interaction into account . The model consists of a system of second order parabolic evolution equations for heat and mass, coupled by nonlinear drift terms, and a state equation which involves a nonlocal interaction potential. The negative entropy turns out to be Lyapunov functional of the system and yields the key estimate for proving global existence and uniqueness results and for analyzing the asymptotic behaviour as time goes to infinity.
Adiabatische Limites wie etwa die Born-Oppenheimer-Approximation lassen sich mit schwachen Konvergenzmethoden herleiten und verstehen. Wir diskutieren die Vor- und Nachteile dieses Zugangs und Erweiterungen, die es erlauben, in nicht adiabatischen Situationen wie Energiekreuzungen der Kodimension 2 sogenannte Landau-Zener-Formeln herzuleiten.
In this talk I will try to describe (part of) an ongoing reasearch with D. Preiss and M. Csörnyei. The starting point is a decomposition theorem for null sets (sets of measure zero) in the plane, which can be obtained as a direct consequence of a geometric version of the Erdös-Szekeres theorem. Then I will outline some applications to problems in Geometric Measure Theory (and not only). If time permits, I will finally address the issue of extending these results to higher dimensions.
We will discuss the asymptotic behaviour of the ground state (principal eigenpair) for a singularly perturbed second order elliptic operator defined on a compact manifold, or in a bounded domain with Dirichlet boundary condition.
It will be shown that the limit behaviour of principal eigenvalue and the logarithmic asymptotics of principal eigenfunction can be described in terms of auxiliary variational problems not involving a parameter.
First we follow the approach of the BCRE model for slow deposition (no inertia, no wind forces) of granular matter under the influence of gravity: We distinguish a standing and a rolling layer with exchange between these layers depending on whether the actual slope exceeds the angle of repose. Then we consider equilibrium configurations, in particular those which can be characterized by a maximum volume property. These can be seen as solutions to certain boundary and obstacle problems for the eikonal equation. The latter can be characterized by variational principles, by Perron methods, and by using transport paths.
The Cauchy-Born rule postulates that when a monatomic crystal is subjected to a small linear displacement of its bounday, then all atoms will follow this displacements. In absence of previous mathematical result, we study the validity of this rule in the model case of a 2D cubic lattice interacting via harmonic springs.
Our main result is that for favourable values of the spring constants and equilibrium spring lengths, the CB rule is actually a theorem.
Simple counterexamples show that for unfavourable spring parameters or large displacements the CB rule fails. Moreover the resulting overestimation of the lattice energy per unit volume by the CB rule can not be cured by convexificaton (let alone quasiconvexification) of the CB energy.
The main tool in the proof is a novel notion of lattice polyconvexity which allows to overcome the difficulty that the elastic energy as a function of atomic positions can never be convex, due to frame-indifference.
We discuss the (somewhat surprising) regularizing effects on BV functions of the symmetric rearrangement operator. These properties are applied to the discussion of the equality cases in the Pólya-Szegö type inequalities.
Quasiconformal mappings are closely connected to elliptic PDE's. In the plane the Beltrami operators provide an efficient and systematic tool to study and apply these relations. I will describe the basic aspects of Beltrami operators with recent applications to PDE's. The applications include solution to a question of Pucci on equations of nondivergence type, QC Young measures, regularity of isotropic equations (work of Faraco). If time permits also recent applications to inverse problems are discussed.
Stefan problems are widely used to model the freezing/melting process of water/ice. In this talk a general existence and uniqeness result of classical solutions for a class of Stefan problems with Gibbs-Thomson correction in arbitray space dimensions is provided. In addition, it will be shown that the moving boundary depends analytically on the temporal and spatial variables. Of crucial importance for the analysis is the property of maximal Lp-regularity for the linearized problem, which is based on the Dore-Venni theorem.
A Feynman-Kac type representation formula for the weak solutions of parabolic geometric flows will be discussed. These flows are initial value problems for sets in the Euclidean space and the evolution rule is a possibly nonlinear function of the curvature. Typical examples are the general codimension mean curvature flow, inverse mean curvature flow and the smooth crystalline flow of Gurtin. This representation characterizes the solution at time t as the set of points in the ambient space from which a carefully chosen stochastic process reaches the initial set with probability one. This is a stochastic controllability problem. In the case of mean curvature flow the stochastic process is a Brownian motion living on an N-d dimensional plane. Here the orientation of the plane is a control parameter we get to choose and d is the codimension and N is the dimension of the ambient space.
After a brief introduction to Ito calculus, the representation will be proved for smooth flows. This is joint work with Nizar Touzi from Paris.
We prove that the two dimensional free magnetic Schrödinger operator, with a fixed constant magnetic field and Dirichlet boundary conditions on a planar domain with a given area, attains its smallest possible eigenvalue if the domain is a disk. This generalizes the classical Faber-Krahn inequality for magnetic fields. The result is used to determine the low energy asymptotic behaviour of the integrated density of states of the magnetic Schrödinger operator with a Poissonian random potential.
The problem of global existence for the solutions of the multidimensional Navier-Stokes equations for the compressible viscous fluids has generated great interest among mathematicians. Starting from the pioneering papers of D. Gaffi [1] and J. Serrin [2] the basic initial boundary value problems were formulated and the uniqueness of the classic solutions was established. The first existence result was obtained by J. Nash [3] for the classic solution of the Cauchy problem on a sufficiently small time interval. For mixed problems local solvability in time was proved by V.A. Solonnikov [4] in the case of a baratropic fluid and by A. Tani [5] in the case of heat-conducting fluid.
We study the question of global solvability of an initial boundary value problem for the Navier-Stokes equations of a compressible viscous heat-conducting fluid in function spaces in which there is local solvability. Examples are given for the solution destructs in finite time: the solution loses smoothness.
1. D.Graffi, J.Rat.Mech.Anal.2 (1953), 99-1062. J. Serrin, Arch.Rat.Mech.Anal. 3 (1959), 271-2883. J. Nash, Bull.Soc.Math. France 90 (1962), 487-4974. V. A. Solonnikov, Engl. transl. in J.Soviet.Math. 14 (1980), no. 25. A. Tani, Publ.Res.Inst.Math.Sci. 13 (1977/78), 193-253
The Cauchy singular integral operator, S, is one of the main actors in the theory of Fourier convolutions, Toeplitz operators, Riemann-Hilbert problems, Wiener-Hopf and singular integral equations. While the boundedness of S has been studied for many decades, final results on the spectrum of S were obtained only in recent times. During the last few years, it was discovered that there is a surprising and undreamt-of metamorphosis of the (local) spectra of S from circular arcs via horns and logarithmic double-spirals to so-called logarithmic leaves with a halo. The talk is a survey of this fascinating development. Technical details will be omitted, but many beautiful pictures of local spectra will be shown. It will be a feast for the eyes.
We review some recent results on the existence of three-dimensional global in time compressible isentropic flows. In particular, positive existence results are given for the physically relevant values of the adiabatic exponent gamma > 3/2.
In a joint work with Tristan Riviere, we have studied a family of energy-functionals related to micromagnetics, in dimension 2, containing an "exchange-energy" term and a "demagnetizing-energy" term, when the constant in front of the exchange-energy tends to 0. We showed a compactness result for families of uniformly bounded energy. Such configurations converge to unit-valued divergence-free vector fields that are tangent to the boundary of the domain (such configurations have typically line singularities), and we extract a limiting energy (sort of Gamma-limit) which penalizes the jumps of the limiting configuration. We also give a kinetic formulation of the problem which allows to get an entropy-type condition for almost-minimizing families.
In many applications such as acoustics or geophysics, fluid models are written in terms of small dimensionless parameters controlling diffusion or wave propagation. The asymptotic regime of vanishing parameters sometimes generates waves and small scale flow near boundaries called boundary layers. We present recent results based upon partial differential equations techniques on nonlinear stability or instability of boundary layers applied to Ekman layers which arise in the small Rossby number limit of the rotating Navier-Stokes equations relevant to the earth regime. We also study the convergence of the low Mach number limit of the compressible Navier-Stokes equations with boundaries.
We study the interface between water and air in a porous medium. In order to simplify the geometry we assume the medium to consist of a periodic array of cylinders of size and distance $O(\eps)$. In between the cylinders there is fluid (lower part) or air (upper part). We apply the method of two-scale convergence to find limiting equations. Inside the (unknown) fluid-domain this is the Darcy-law. Along the interface we can derive a nonlinear pressure-condition that determines the position of the free boundary. We have to deal with the fact that the contact-angle equation introduces a capillary pressure of the order $1/\eps$.
Linear and nonlinear evolution equations modeling elastic systems with dissipation effects are considered, the investigation of the asymptotic behavior of solutions as time tends to infinity being the main topic.
First we give a survey on results in linear and nonlinear thermoelasticity - thermoelastic bars, membranes, contact problems. Second we report on recent research on magneto-thermo-elastic systems.
We consider the relativistic electron-positron field interacting with itself via the Coulomb potential defined with the physically motivated, positive, density-density quartic interaction. The more usual normal-ordered Hamiltonian differs from the bare Hamiltonian by a quadratic term and, by choosing the normal ordering in a suitable, self-consistent manner, the quadratic term can be seen to be equivalent to a renormalization of the Dirac operator. Formally, this amounts to a Bogolubov-Valatin transformation, but in reality it is non-perturbative, for it leads to an inequivalent, fine-structure dependent representation of the canonical anticommutation relations. This non-perturbative redefinition of the electron/positron states can be interpreted as a mass, wave-function and charge renormalization, among other possibilities, but the main point is that a non-perturbative definition of normal ordering might be a useful starting point for developing a consistent quantum electrodynamics.
Atmospheric flow theory relies to a large extent on scale analyses and perturbation methods. These analyses explore the smallness of various non-dimensional characteristic numbers, such as the Froude-, Mach-, and Rossby-numbers and various length and time scale ratios. Notably the Mach number, being the ratio of a characteristic flow velocity and a typical sound speed, is independent of the considered length and time scales, and it is almost everywhere small within the atmosphere. This observation suggests the following approach towards deriving a hierarchy of simplified asymptotic model equations for atmospheric flows:
1. Use the Mach number, M, as the basic asymptotic expansion parameter. 2. Consider flows on increasingly larger scales, beginning with fluid motion on scales of a few meters, and ending with synoptic scale or even global atmospheric dynamics. 3. As the considered length scale increases, the above-mentioned scale dependent characteristic numbers will decrease. Whenever one of these numbers, say K, becomes small and justifies an estimate K = M**a as M -> 0, a new distinguished limit can be introduced and the associated flow regime can be explored through low Mach number asymptotics. This presentation will first show that this procedure allows one to recover many, if not most, of the well-established simplified model equations for atmospheric flow dynamics in a unified fashion. By introducing multiple scales expansions in space and/or time, we reveal additional interactions between phenomena acting on different length and time scales. Implications of these results for the construction of efficient and accurate numerical integration schemes for atmospheric flow dynamics will be summarized in the end.
Most of the theory of the Boltzmann equation has been built upon 'Grad's cut-off' assumption, i.e that the kinetic cross-section appearing in Boltzmann's collision operator should be locally integrable. This assumption is however true only for particles interacting via finite-range forces. The case of long-range interactions has been the object of a lot of interest in the past few years, due to the complete change in the structure of the equation, and in the qualitative behavior of solutions. In this talk, I present the extension of the DiPerna-Lions theory to the case of long-range interactions, together with the proof of a conjecture by Lions, about regularizing properties of the Boltzmann equation without cut-off. Parts of the work were done in collaboration with Desvillettes, Alexandre, Wennberg.
Entropy methods are well-known in the analysis of hyperbolic equations. Recently, so-called entropy dissipation methods have been introduced for the analysis of the qualitative behavior of solutions to parabolic equations. The entropy method gives additional a priori estimates for the solutions of the corresponding equations.
In this talk we present by examples the principal ideas of these entropy methods. In particular we consider strongly nonlinear parabolic systems and higher-order parabolic equations. For these problems maximum principles are not available. The entropy method will be essential to prove (i) existence of solutions, (ii) explicit long-time decay rates of solutions, (iii) stability of certain numerical schemes, (iv) asymptotic limits, and (v) uniqueness of solutions.
We present some new results on the strong solvability of the Navier-Stokes equations if the initial velocity belongs to suitable spaces of distributions. In particular, we present an analogue to a recent result of Cannone for the case of a bounded domain and discuss uniqueness questions.
How does the range of the gradient of a lipschitz map look like? Here we consider finite sets of matrices as possible candidates and give a new class of examples - gradients without rank-one connections. The sharp estimate of the minimal number of gradients involved is related to properties of quasiconformal mappings, a geometric approach to rank-one convexity and a new, unifying way how to solve partial differential inclusions.
Our purpose is to extend to the framework of Electromagnetism the Monge-Kantorovich theory which originally comes from Continuum Mechanics and has become very popular in the last ten years in the field of nonlinear PDEs, especially because of its connection with the Monge-Ampere equation.
We construct, in the framework of Electromagnetism, an Action (related to the Nambu-Goto Action for extremal surfaces) which is analogous to the Monge-Kantorovich (or Wasserstein) distance. From this Action, we derive a system of PDEs that look like non-linear Maxwell equations. Then, through suitable approximations, we recover not only the linear Maxwell equation, as expected, but, more surprisingly, the complete (pressureless) Euler-Maxwell system.
I will discuss several mathematical problems related to curvature driven flows arised from interfacial motions in crystal growth. Emphasis will be on stochastic effects. Specific examples of such flows include motion by mean curvature and dendritic crystal growth with Gibbs-Thompson condition. In some cases, noise can be shown to give a selection principle to the evolutions in case of non-uniqueness.
It is a well known fact that in the course of their evolution higher organisms rapidly grow to a size where passive diffusion is not enough to provide the tissues with oxygen, nutrients, water and information. Nature has found a way out to this problem by the invention of complex-shaped organs of a filamentary structure, which thoroughly expand into the corresponding organic matrix. The vascular system of vertebrates is a paradigmatic example of such a situation.
While the underlying genetic mechanisms have begun to be elucidated only recently, mathematical models of reaction - diffusion type, accounting for some of the qualitative features observed, were proposed as early as in the 70s. Here we shall describe one such model, which was introduced by Hans Meinhardt. Some of the main features of the corresponding solutions, as for instance the local structure of the expanding filaments and the branching phenomena taking place there, will be presented and discussed by means of matched asymptotic expansions techniques.
One considers the vanishing viscosity method for hyperbolic systems of conservation laws. The main application is the Euler equations of an inviscid, non-conducting gas, approached by the Navier-Stokes equations. We restrict ourselves to a non-characteristic boundary, in particular, the net flow accross it is not zero. In this context, a boundary layer (B.L.) forms, which role is to match the Dirichlet boundary condition with the hyperbolic solution in the interior (I.S). The evolution of the B.L. and of the I.S. are coupled.
The convergence as the dissipation parameters tend to zero was proved in 1-d by Gisclon and Serre (1994) and in any dimension by Grenier and Gues (1998), under the asumption that the amplitude of the B.L. is moderate. We understood that instability might occur for larger B.L.s. The problem has been identified : the linearized IBVP around the B.L. may have eigenvalues with positive real part, which behave like the inverse of dissipation.
The search of such eigenvalues is a difficult task, which can be partly solved using an Evans function. In some cases of physical interest, one exhibits unstable B.L.s. Typically, such B.L.s are close to strong steady viscous shock waves.
The Hodge theorem provides a canonical choice of representatives for DeRham cohomology. His original theory applies to compact manifolds and relies on resolvent estimates which seem to require compactness. In this lecture I will explain how his theory can be extended to a class of non-compact manifolds which are sufficiently trivial at infinity. The basic analytic tool is an estimate which holds when Brownian motion is subjected to a very strong centripetal force field.
In this talk, we will discuss a method to construct nonconvex free energies which are invariant under a (usually discrete) symmetry group and vary with temperature. Applying methods of group and representation theory, this approach yields a formal description of all $C^\infty$ potentials. Using these techniques and geometric arguments, the construction of concrete potentials is comparatively easy. As an example, the cubic-tetragonal phase transition will be discussed; additionally, the problem of growth conditions at infinity will be addressed.
In dem Vortrag wird eine Klasse (räumlich) wechselwirkender Diffusionen aus der Populationsgenetik unter der Perspektive des Langzeitverhaltens studiert. Besonderes Gewicht liegt auf Systemen, die sich im zufälligen Medium entwickeln. Für spezielle Systeme ist eine detaillierte Analyse durch explizite Rechnungen möglich. Durch Kopplungs- und Vergleichstechniken für die zugehörigen Feller-Halbgruppen lassen sich hieraus Resultate in größeren Modellklassen erzielen. Der Grund für diese Universalität lässt sich durch Renormalisierungstechniken aufdecken, die den stochastischen Systemen Orbits nichtlinearer Abbildungen auf Funktionenräumen zuordnen.
Travelling waves in periodically perforated domains are shown to exist for large classes of reaction-diffusion systems, provided the homogenized equation admits a nondegenerate travelling wave. This can be applied e.g. to a single equation with bistable nonlinearity and to bistable monotone systems. The proof uses the implicit function theorem of a suitably transformed problem in the space H1. Furthermore corrector type estimates are given for the wave profile and the wave velocity.
The Kohn-Sham system has been well established as an instrument for calculations in several branches of physics and chemistry since many years (Nobel Prize 1998). Astonishingly, it had not been investigated mathematically as concerns questions as existence of solutions and their regularity uniqueness strict justification of numerical procedures before the last years. The author presents results on this item which he obtained together with H.-C. Kaiser in 96/97.
We show that if the Ginzburg-Landau energy is uniformly bounded for a sequence of functions uε as ε → 0, then the Jacobians {Juε} are precompact in a ppropriate weak topologies. We further show that any limiting measure must be rectifiable. These results have potential applications in problems a variety of problems, including for example questions involving dynamics of vortex filaments in superfluids. (joint work with Mete Soner)
In the modelling of electrons as quantum mechanical particles several types of NLS arise. The first class are "weakly nonlinear" models such as the Schroedinger-Poisson equation. Relativistic extensions of this model are given by the Dirac-Maxwell system and selfconsistent Pauli equations. Another nonlinearity arises in the context of the Hartree-Fock equations.
It can be approximated by a local function of the density similar to the focusing case of the cubic NLS. We present the models and the open mathematical problems as well as recent progress e.g. on the limit from Schroedinger- Poisson in a crystal to the semiclassical equations of solid state physics using a new variant of Wigner transforms.
The subject of this communication is the theory of solvability of boundary value problems for the Stokes equations with the velocities or stresses prescribed at the boundary in the spaces of functions whose derivatives are Hölder continuous with respect to spatial variables and time. A special emphasis will be given to coercive estimates in these spaces and to their applications.
We consider three mathematical models describing behaviour of nonisothermal liquid in microgravity conditions. First of them is alternative with respect to the classical Oberbeck-Boussinesq model of convection. It deals with a liquid which density is the function of temperature only but the velocity vector is no longer solenoidal. Second one is the linear model of convective flow in a weakly compressible liquid. Third model describes the motion of an emulsion or a gas-liquid mixture exposed to the thermocapillary forces and microacceleration.
Some "a priori" estimates for minimizers of functionals with free discontinuities can be improved in such a way to give the initial assumptions which make some bootstrap regularity arguments start. Such arguments have been employed so far thanks to generic properties of the singular set and have led to partial regularity results. Their systematic use allows stability methods which lead to prove new properties of the singular set, in particular for the case of two dimentions.
We study the incompressible Navier-Stokes equations on and prove existence and uniqueness of a solution u in with provided the solution v to the heat equation with the same initial data satisfies
This condition on the initial data is local in space and frequency. It extends Kato's wellposedness result in since .The function space is closely related to BMO: if there exists a vector field with . This is joint work with D. Tataru.
We consider a nonlinear transport equation as a hyperbolic generalisation of the well-known reaction diffusion equation. We show the existence of strictly monotone travelling fronts for the three main types of the nonlinearity: the positive source term, the combustion law, and the bistable case.
In the first case there is a whole interval of possible speeds containing its strictly positive minimum. For subtangential nonlinearities we give an explicit expression for the minimal wave speed.
In this talk, we will discuss two methods (a global one and a local one), developped in collaboration with P.L.Lions, to handle the compressible-incompressible limit when the Mach number goes to zero. The difficulty comes from the pesence of high oscillations in time (acoustic waves). We will also give some application to the case of vanishing viscosities and also to the Boltzmann equation.
This work was done jointly with Robert Neel. We consider the problem of determining the surface interface of fluid partly filling a semi-infinite capillary tube of general section D and closed at one end, in the absence of gravity. It is known that the formal equations for a surface interface as a graph covering the base do not in general admit regular solutions. In the present work it is shown that whenever smooth solutions fail to exist there will nevertheless exist a solution in a singular sense suggested by physical intuition. These surfaces are smooth graphs with constant mean curvature H, over subsets of the base domain D that are bounded within D by subarcs of semicircles C of radius 1/2H and meeting the boundary of D in the prescribed contact angle. As C is approached within the subdomain of regularity, the solution surface is asymptotic to the vertical cylinder over C. Surfaces with that behavior have been observed physically.
In some configurations of particular interest, the procedure leads to unique determination of such a "C-singular" solution. However, uniqueness cannot in general be expected, as is shown by example. Further examples show a) that C-singular solutions may appear also when smooth solutions exist, and b) they may fail to occur in that case, depending on the particular geometry.
We construct cmc immersions of annular type spanning a pair of oriented rectifiable Jordan curves by solving an appropriate volume-constrained Douglas Problem. We focus on the case where the boundary curves are circles. In particular, we show that solutions of large volume become spherical. There is a bubbling off phenomena.
Let Γt be a compact embedded hypersurface \(\mathbb(R)^{n+1}\) which moves with speed determined at each point by a function F(k1,...,kn,t) of its principal curvatures, for 0≥t≥T. We assume the problem is degenerate parabolic, that is, that F(.,t) is nondecreasing in each of the principal curvatures k1,...,kn We shall show that for t > 0 the hypersurface Γt satisfies local a priori Lipschitz bounds outside of a convex set determined by Γ0 and lying inside its convex hull. As a consequence, if Γt expands to reach points further and further away from Γ0 then Γt converges to a round sphere after rescaling. This is proved without estimates on curvature, in contrast to earlier proofs of asymptotic roundness, which typically require strict parabolicity. Our method is the parabolic analogue of Aleksandrov's method of moving planes.Aleksandrov's reflection method is also extended to treat generalized solutions of this evolution problem, that is, level sets of viscosity solutions to the corresponding geometric PDE. These generalized solutions have recently been shown in certain cases to develop a nonempty interior after the evolving hypersurface collides with itself or develops singularities. We shall prove that the same local Lipschitz bounds as in the embedded-hypersurface case hold for the inner and outer boundaries of such a "fattened" level set Γt.As an application, we give some new results about 1/H flow for nonconvex hypersurfaces, which was recently investigated by Huisken and Ilmanen. Our solutions are limits of viscosity solutions, and are therefore solutions of a local problem, in \(\mathbb(R)^{n+1}\) only. In contrast, the evolving hypersurfaces given by Huisken and Ilmanen, which were used to prove a version of the Penrose conjecture, are solutions of a non-local problem, valid in general asymptotically flat Riemannian manifolds.This is joint work with Bennett Chow.
We consider bounded entire solutions of semilinear elliptic equations of the form Δ u = F'(u) in R3, satisfying δx3 u > 0 in R3 and u(x1,x2,x3) → ± 1 as x3 → ± ∞. We do not assume these limits to be uniform with respect to (x1,x2). Under the hypothesis that F ≥ min{F(1),F(-1)} in [-1,1] , we prove that the level sets of u are planes. That is, u is a function of one variable only. This establishes in dimension three a conjecture formulated by De Giorgi in 1978. In dimension two, this conjecture was recently proved by Ghoussoub and Gui, and similar results were obtained by Berestycki, Caffarelli and Nirenberg. The conjecture remains open in dimension n ≥ 4.
AMS Classification: 35B05, 35B40, 35B45, 35J60
Keywords: Semilinear elliptic PDE, Symmetry and monotonicity properties, Energy estimates, Liouville theorems
We construct the simplest solution of the Einstein equations that incorporates a shock-wave into a standard Friedmann-Robertson Walker metric whose equation of state accounts for the Hubble constant and the microwave background radiation temperature. This produces a new solution to the Einstein equations from which we are able to derive estimataes for the shock position at present time. We show that the distance from the shock-wave to the center of the explosion at present time is comparable to the Hubble distance. We are motivated by the idea that the expansion of the universe as measured by the Hubble constant might be accounted for by an event more similar to a classical explosion than by the well-accepted scenario of the Big-Bang.
We present a theory of bifurcation from symmetry, based on topological techniques (Conley Index) and group representations. We apply the theory to equations of the form: Laplacian(u)+f(u)=0, on balls in $R^n$ with general homogenous boundary conditions, and we prove the existence of asymmetric (non-radial) solutions.
We discuss a recently derived model for the propagation of waves in shallow water. The equation is integrable and its solitary waves are solitons. The only way a singularity can arise in a classical solution is in the form of wave-breaking. The nonlinear PDE can be viewed as geodesic flow on the manifold of diffeomorphisms of the line. This geometric interpretation is very useful in dealing with qualitative aspects of the equation.
Abstract: We will define a notion of geometric integrability for non-linear second-order hyperbolic partial differential equations in the plane which is based on the behaviour of the characteristic systems. We will then see how to characterize the geometrically integrable equations in terms of their conservation laws through the cohomology of the corresponding constrained variational bi-complex.
Let M be a graph embedded in the plane. The lace around M is the open set Oε of thickness 2ε around the edges of M, and smoothed close to the vertices of M. The details of the smoothing will be proved to be irrelevant. The unknowns in the two-dimensional Ginzburg-Landau functional are the vector potential and the complex order parameter. As ε tends to 0, it is possible to extract from any sequence of minimizers a converging subsequence whose limit is a minimizer of a one-dimensional Ginzburg-Landau functional on M. The only unknown in this functional is a complex order parameter. A further reduction leads to a new functional depending only on a realvalued function and n integers, n being the number of independent cycles of the graph, or equivalently the number of holes in R2 \ M.
Quasiharmonic fields are a young area of research. It links with the conjecture about "rank-one-convexity implies quasiconvexity", but the focus is on applications to linear elliptic PDEs with discontinuous coefficients (nonuniformly elliptic).
In this talk we discuss the dynamics of one-dimensional lattices with nearest neighbor interactions, initially at rest, and which are driven from one end by a particle τ0. The driver, τ0, is assumed to undergo a prescribed motion, where a, ε, γ are real constants and h(.) has period 2π. We describe the numerically observed behaviour (shock and rarefaction phenomena for ε = 0, generation of multi-phase travelling waves for ε ≠ 0) and present corresponding analytical results.
Special emphasis is given to the integrable model F(x) = ετ (Toda lattice). In this particular case one can rigorously derive the long-time asymptotics for a large class of initial value problems using the Inverse Scattering Transform (IST) method. Hereby we formulate the IST as a matrix-valued Riemann-Hilbert problem. Such Riemann-Hilbert problems have recently been used to prove asymptotic results in a variety of different fields, such as integrable systems, statistical mechanics, combinatorics and orthogonal polynomials.
We consider the laminar viscous channel flow, with the lateral surface of the channel containing surface irregularities. It is supposed that a uniform pressure gradient is maintained in the longitudinal direction in the channel. After studying the corresponding boundary layers, we obtain rigorously the Navier friction condition. It is valid when the size and the amplitude of the imperfections tend to zero. Furthermore, the coefficient in the law is determined through an auxiliary boundary-layer type problem and the tangential drag force and the effective mass flow are determined at order corresponding to the characteristic size of the impurities at the power 3/2.
Variants of the time dependent (parabolic) harmonic map equation arise in several applications such as magnetostriction and fibre spinning. In the present lecture I shall discuss some relationships between recently discovered singularities, modelling questions, the nonuniqueness of initial-boundary value problems, and the construction of solutions.
I will begin by providing some numerical results for the semiclassical limit of the focusing NLS equation. I will then discuss the dispersionless KdV analysis of Lax and Levermore. I will continue by introducing a recent method for solving asymptotic problems for integrable equations (making use of the Riemann-Hilbert factorization formulation of the associated inverse spectral problem). Finally I will apply this method to show how rigorous results for the semiclassical NLS can be obtained.
In recent years considerable activity was directed to the issue of stability in the case of matter interacting with an electromagnetic field. We shall review the results which have been established by various groups, in different settings: relativistic or non-relativistic matter, classical or quantized electromagnetic fields. Common to all of them is the fact that electrons interact with the field both through their charges and the magnetic moments associated to their spin. Stability of non-relativistic matter in presence of magnetic fields requires that Zα2 (where Z is the largest nuclear charge in the system) as well as the fine structure constant α itself, do not exceed some critical value. If one imposes an ultraviolet cutoff to the field, as it occurs in unrenormalized quantum electrodynamics, then stability no longer implies a bound on α, Zα2. An important tool is given by Lieb-Thirring type inequalities for the sum of the eigenvalues of a one-particle Pauli operator with an arbitrary inhomogeneous magnetic field.
We consider the Extended Fisher-Kolmogorov equation where is a double-well potential and γ>0. For γ=0 this is the ordinary Allen-Cahn equation. The equation for the stationary waves is the ordinary differential equation with appropriate boundary conditions. In this talk, we present estimates on the second derivatives of solutions of (2) which enable to prove that bumps of these solutions have to have a minimal size, hence cannot accumulate. As main result, we prove that the area functional is the Γ-Limit of which is a Ljapunov functional of (1). This is joint work with Danielle Hilhorst and Lambert A. Peletier.
We consider long time asymptotics and attractors of all finite energy solutions to Hamiltonian nonlinear wave equations in the whole space. We prove long-time convergence to stationary states for the solutions in a Fréchet topology defined by local energy seminorms. This means that the set of stationary states is a point attractor for the systems in this topology.
The convergence in the global norm is impossible due to energy conservation. For translation-invariant systems the convergence implies the soliton-like asymptotics in some global norms.
The convergence is established for one-dimensional nonlinear wave equations with nonlinear terms concentrated at a single point /1/, at a finite set of points, /2/, and at a finite segment, for three-dimensional scalar wave equation coupled to a particle, /3/ and for Maxwell-Lorentz system with a charge, /4/. We establish the soliton-like asymptotics for finite energy solutions to the translation-invariant three-dimensional scalar wave equation coupled to a particle, /5/.
Liapunov-type criterion is established for asymptotic stability in the Fréchet topology of the stationary states of general nD nonlinear wave equations and systems, /6/.
The investigation is inspired by an outstanding role of the stationary states in many phenomena described by nonlinear Hamiltonian wave equations: N. Bohr's postulate on transitions to stationary states in quantum systems, E. Schrödinger's wave mechanics, L. de Broglie's wave-particle duality and radiative damping in classical electrodynamics, /7/.
References A. Komech, On stabilization of string-nonlinear oscillator interaction, J. Math. Anal. Appl., v. 196 (1995), 384-409.A. Komech, On the stabilization of string-oscillators interaction, Russian Journal of Mathematical Physics, 3 (1995), 227-248. A. Komech, H. Spohn and M. Kunze, Long-time asymptotics for a classical particle interacting with a scalar wave field, Comm. Partial Diff. Equs. 22 (1997), 307-335. A. Komech, H. Spohn, Long-time asymptotics for the coupled Maxwell-Lorentz equations, submitted to Comm. Partial Diff. Equs. A. Komech, H. Spohn, Soliton-like asymptotics for a classical particle interacting with a scalar wave field, Nonlinear Analysis 33 (1998), 13-24. A. Komech, B. R. Vainberg, On asymptotic stability of stationary solutions to nonlinear wave and Klein-Gordon equations, Arch. Rat. Mech. Anal. 134 (1996), 227-248.A. Komech, On transitions to stationary states in Hamiltonian nonlinear wave equations, Phys. Letters A 241 (1998), 311-323.
A general critical point theory can be developed for continuous functions defined on a metric space. Also some classes of lower semicontinuous functions can be approached. A wide area of application is constituted by some quasilinear elliptic equations with natural growth conditions, which are Euler equations of continuous but not locally Lipschitz functionals.
The bilinear Hilbert transform is defined by It has been shown recently that this bilinear operator satisfies estimates of the type for certain p, q, and r. This is the analogue of the classical Riesz-Kolmogorow theorem for the (linear) Hilbert transform. The purpose of this talk is to discuss these and related results. A main tool is the decomposition of functions into wave packets, i.e., well localized functions whose Fourier transform is also well localized. Such decompositions are frequently used in applied mathematics such as signal or image processing.
We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. In a first part we we consider boundary control problems with either Dirichlet or Neumann conditions. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. It is shown that a recently developed interior point method is able to solve theses problems even for high discretizations. Several numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang-bang and singular controls. The necessary conditions of optimality are checked numerically in the presence of active control and state constraints.
This is joint work with H. Maurer, Universitaet Muenster.
We study long time behavior of solutions to the quasi-steady approximation of the equation of viscoelasticity with capillarity. We consider the equation with a number of boundary conditions including such that the set of equilibria is necessarily at least one-dimensional. We prove that any solution converges to an equilibrium point. We admit some kind of initial data with infinite energy. We use tools based on the assumption that the nonlinear term is real analytic and our observation that the equation in question generates a gradient flow in $H^{-1}$.
We also remark that the same method applies (with changes) to Cahn-Hilliard equation.
Some analytical, numerical and experimental techniques arising in modelling small loops of DNA will be described. The experiments probe the mechanical properties of DNA, the mathematics involves symmetry breaking and stability exchange in isoperimetric calculus of variations problems, and the computations involve parameter continuation techniques and visualization.
Using calculus of variations techniques, it is shown that if un ∈ W1,N(Ω;RN) converge to a function u in L1(Ω;RN), where Ω is an open, bounded subset of RN, if the sequence of all minors {M(𝛻un)} is equi-bounded in L1, and if det𝛻un converge to a function f weakly in L1(Ω), then f = det𝛻u a.e. 𝜘 ∈ Ω.
This result was previously obtained by Giaquinta, Modica and Soucek using tools from Geometric Measure Theory, and it is sharp. In particular, for all q ≥ 1, 1 ≤ p < N - 1, for all f ∈ Lq(Ω), and for every u ∈ W1,p(Ω;RN)
This work in collaboration with Jan Malý was initiated during his visit to the Max Planck Institute in March-April 1998.
Phasenumwandlungen sind häufig mit Hysterese-Effekten verbunden, vor allem dann, wenn die Umwandlungsprozesse zyklisch verlaufen. Da Hysterese-Effekte durch die derzeit üblichen Phasenfeldsysteme zur Modellierung von Phasenumwandlungen nur sehr eingeschränkt erfaßt werden können, sind diese Modelle für konkrete Anwendungen häufig nicht befriedigend.
Im Vortrag wird ein neuer Ansatz vorgeschlagen, der auf einer Verwendung der sogenannten "Hysterese-Operatoren" beruht und den Vorteil hat, die Hysterese-Effekte direkt auf der makroskopisch-phänomenologischen Skala in das Modell zu integrieren. Der Ansatz hat den Vorteil, dass die Evolution des Systems stets thermodynamisch konsistent verläuft und dass im Gegensatz zu Modellen, die nichtkonvexe Potentiale verwenden, niemals thermodynamisch instabile Zweige durchlaufen werden können. In mathematischer Hinsicht wird man auf Systeme partieller Differentialgleichungen geführt, bei denen Nichtlinearitäten vom Hysterese-Typ in den höchsten Ableitungen auftreten. Für eine Klasse solcher Systeme, in der insbesondere hysteretische Analoga der Phasenfeldmodelle von Caginalp und Penrose-Fife enthalten sind, werden Wohldefiniertheit und thermodynamische Konsistenz gezeigt.
Some regularity results are presented for local minimizers of integral functionals satisfying non standard growth assumptions, i.e., such that the growth exponent q and the ellipticity exponent p are different in the inequality Both the homogeneous-anisotropic and the inhomogeneous-isotropic cases are considered.
A mean-field limit is derived for a spin-flip dynamics with Kac-interaction and bounded flip-intensities. This is done by constructing the interacting spins from a family of independent spins flipping at uniform rate by multiparameter time changes. The convergence of these time changes in probability uniformly on small time intervals to the time changes defining the so-called mean-field-process is shown and a convergence theorem for the averaged process towards a deterministic limit is derived.
In 1971 J.M. Anderson conjectured that for any conformal map 𝜑 in the unit disc there exists β,0 ≤ β ≤ 2π such that More recently this problem has been posed in the works of N. Makarov, Ch. Pommerenke and D. Gnuschke - Ch. Pommerenke. The purpose of the present talk is to prove Anderson's conjecture. This will be done by showing the following theorem about the associated Bloch function b = log|𝜑'|.
This talk will review the analogies and differences between macroscopic models for fluid and superconducting vortices and dislocations in crystals, each being regarded as a "free" line singularity of solutions of a partial differential equation. A systematic derivation will be given of the dynamics of the line singularity in each case as well as some of the implications for homogenised models in which many singularities are present. A difficulty arises with dislocations where it appears that the discrete structure of the core needs to be taken into account.
The variational limit of discrete models of atoms subject to nearest-neighbour or multiple-neighbour non-linear interactions describes continua allowing for softening and fracture.
The potential of wavelets as a discretization tool hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted sequence norms of wavelet coefficients as well as on certain cancellation properties. Both features are crucial for the construction of optimal preconditioners, for matrix compression and the corresponding sparse representations of functions and operators and the disign and analysis of adaptive solvers. So far the availability of such bases is confined to very simple domain geometries.
This talk is concerned with concepts that aim at expanding the applicability of current wavelets schemes. The key is to construct wavelet bases with the desired properties on manifolds which can be represented as the disjoint union of smooth parametric images of the standard cube.
The approach is based on the characterization of function spaces over such a manifold in terms of product spaces where each factor is a corresponding local function space subject to certain boundary conditions. Wavelet bases for each factor can be obtained as parametric liftings from bases on the standard cube satisfying appropriate boundary conditions. The use of such bases for the discretization of operator equations leads in a natural way to a conceptually new domain decomposition method whose convergence properties are well understood also for operators of nonpositive order.
Es wird die klassische und quantenmechanische Bewegung eines Teilchens in einem Potential mit m anziehenden Coulombsingularitäten untersucht.
Im Gegensatz zu den integrablen Fällen m=1 (Newton) und m=2 (Jacobi) ist die Streuung an diesem Potential für größere m irregulär und wird durch eine Cantormenge g e b u n d e n e r Bahnen positiver Energie bestimmt. Diese Irregularität ist nur im quantenmechanischen, nicht im klassischen differentiellen Streuquerschnitt sichtbar.
Die Bewegung eines klassischen Teilchens hoher Energie in einem periodischen Potential (Kristall) mit Coulombsingularitäten ist diffusiv, die des quantenmechanischen Teilchens ballistisch. Im semiklassischen Limes geht dessen Geschwindigkeit aber gegen Null.
We derive a system of equations, related to the well-known Navier-Stokes equations and Allen-Cahn equation, describing a gas-liquid fluid. We show the consistency of the model with the 2nd law of thermodymics. Furthermore, some numerical calculations in two space dimensions are pres ented.
Der Erfolg von Wavelet-Methoden bei der Anwendung in der Signal- und Bildverarbeitung hängt entscheidend von drei Kriterien ab:
Erstens muss das zugrundeliegende Problem eine Multi-Skalen-Struktur besitzen, das heißt, das zu untersuchende Objekt (Signal, Bild, Operator) muss räumlich lokalisierte Strukturen unterschiedlicher Größenordnung aufweisen.
Zweitens muss ein Wavelet ausgewählt oder konstruiert werden, das den mathematischen Eigenschaften des Problems angepaßt ist.
Letztendlich müssen passende Verfahren zur Auswertung der entstehenden Wavelet-Transformationen entwickelt werden.
In diesem Vortrag soll dies an zwei realen Beispielen (BMW-Rolls Royce, Klimarechenzentrum Hamburg) demonstriert werden. Den Abschluss bildet eine Anwendung auf ein nichtlineares inverses Problem.
