We discuss the transition from the well developed modeling and analysis of passive suspensions to active suspensions (namely, bio-suspensions). Modeling of bacterial suspensions and, more generally, of suspensions of active microparticles has recently become an increasingly active area of research. The focus of our work is on the development and analysis of a mathematical PDE model for the multiscale problem of bacterial suspensions. In particular, we discuss recent results on the effective viscosity of dilute bacterial suspensions (with Aronson, Berlyand and Karpeev).
We study the probability of the motion of an interface between two stable phases of a ferromagnetic system from an initial to a final position within fixed time. We work with a stochastic microscopic system of Ising spins with Kac interaction evolving in time according to the Glauber (non-conservative) dynamics. We derive the cost functional penalizing all possible transitions and we minimize it to find the most probable profile which corresponds to the motion of the interface in the macroscopic scale. This is joint work with N. Dirr and G. Manzi.
Phototransduction is the process by which light, captured by a visual pigment molecule in a photoreceptor cell, generates an electric response. This response is caused by a change of concentrations of certain chemical substances in the cytoplasm of the outer rod segment in the retina of vertebrates.
The basic ideas and results of a work by Andreucci, DiBenedetto and Bisegna are presented. This contains a diffusion model of these substances with suitable Neumann- and Dirichlet boundary data. The complex shape of the outer rod segment leads to a homogenization ansatz with a simultaneous concentration of capacity on the thin surface-like lateral border.
Electrons on a two-dimensional surface subjected to a magnetic field display a curious condensed matter effect known as the quantum Hall effect (QHE). In 1983, R. B. Laughlin proposed a wave function which is now widely used as an approximate ground state for QHE systems at certain densities. In this talk I will present some rigorous results on the thermodynamic limit of Laughlin's function when it is adapted to a cylinder geometry.
This talk is concerned with the asymptotic behaviour of eigenfunctions related to the Schroedinger problem or the Helmholtz equation. After illuminating the physical background, basic properties of the eigenfunctions are derived. The main result is an exponential decay estimate for perturbations of the eigenfunctions caused by a local potential. A one-dimensional example illustrates typical properties of the eigenfunctions as well as limits of the theory.
(Zen)trifugalbeschleunigtes Zell(ku)ltursystem (ZENKU) or centrifugally accelerated cell culture system in English is a device used to culture 3D tissues from stem cells. We investigate the fluid dynamics of such a device experimentally as well as through theoretical modelling. Using this approach we which to better understand the physics involved in tissue formation when cultured in this system.
In the first part of this talk, I will present the definitions and properties of accretive and monotone operators, and discuss some key results developed in the past few decades regarding the solvability of monotone type deterministic PDEs.
Later I will elaborately show an interesting technique (Generalized Minty-Browder Technique) which by exploiting the monotonicity argument, enables us to prove the solvability(strong solution) even in the stochastic case, and avoids the classical compactness method.
We study the role of defects in the quasistatic evolution of a martensitic phase boundary. Martensitic phase transformations involve a change in shape of the underlying crystal, and thus the propagation of the phase boundary is accompanied by an evolving mechanical stress and strain field. This gives rise to a nonlocal free boundary problem, one where the evolution of the free boundary is coupled to an elliptic partial differential equation. Often, real materials contain defects which can potentially pin the interface and contribute to the hysteresis, and understanding this is the goal of this work. We present a mathematical model and a proof of existence in the sense of sets of finite perimeter. We then present numerical simulations and analysis of both the fully nonlocal problem as well as a linearized version, and draw conclusions on the role of defects on hysteresis.
The result I will present is the existence of an extension operator for special functions with bounded variation with a careful energy estimate.
The main application is a compactness result for non-coercive functionals consisting of a volume and a surface integral, as those occurring in computer vision and in the mathematical theory of elasticity for brittle materials. More precisely I will focus on the study of the asymptotic behaviour of the Mumford-Shah functional on a periodically perforated domain, as the size of the holes and the periodicity parameter of the structure tend to zero.
A solution of a problem in nonlinear Elastostatics is a minimizer of the elastic energy of the deformation of a body. This minimum is searched for among all deformations satisfying certain boundary conditions and, in addition, they must preserve the orientation and be one-to-one (so that interpenetration of matter does not occur). When one approximates by finite elements a solution, it is important that those approximations also preserve the orientation and are one-to-one. This leads to the problem of approximating a homeomorphism by piecewise linear ones. This is a classic problem in Topology that is well understood when the approximation is done in the supremum norm. The results that I present show that, in dimension 2, this approximation is also possible in the Holder norm when the original homeomorphism is Holder continuous.
This is a joint work with J. C. Bellido.
We show how to derive Effective parabolic PDE for simplest diffusion process in typical non-periodic heterogeneous media such as extravasculature in tissue, or outside of endoplasmic reticulum at intracellular level.
The classical Taylor-Couette problem concerns the motion of a viscous incompressible fluid in the region between two rigid coaxial cylinders, which rotate at constant angular velocities. This is a fundamental example in bifurcation theory and hydrodynamic stability, and has been the subject of over 1500 papers. This lecture treats a generalization of this problem in which the outer cylinder is a deformable (nonlinearly viscoelastic) shell. Dynamical problems for the interaction of fluids with deformable solids undergoing large displacements are notoriously difficult to analyse. This is a rare instance of such an interaction in which the geometry is simple, the physics is interesting, and the behaviour of solutions can be determined by mathematical analysis, without an immediate recourse to numerical computation.
A theoretical model of DNA self-assembly will be presented. For this model a problem is encoded in the molecules in a pot and a solution is represented by a complete complex (a complex that does not contain free sticky ends) of appropriate size.
In most experiments, besides complete complexes, a lot of undesired material (non-complete complexes) also appears. To optimize the initial solution so as to minimize the amount of non-complete complexes at the end one needs to use proper proportion of molecule types. The set of vectors representing these proper proportions is called the ``spectrum" of the pot.
The spectrum is a convex hull with rational vertices. The extreme points of this convex hull help us to determine some of the minimal complete complexes that could appear in the pot. Also, the spectrum help us to classify the pots.
I will present some already proved facts for the spectrum as well as problems that I am currently working on.
Landau-Lifshitz-Gilbert dynamics governs the evolution of magnetization in a ferromagnetic body. It is a damped Hamiltonian system, describing the combined effect of gyromagnetic precession and damping.
Our attention is on ferromagnetically soft thin-film elements. They are used in many divices and have been explored at length experimentally and numarically. Since for thin films the aspect ratio is small, it it natural to seek a two-dimentional effective equation, expressing the asymptotic dynamics in the limit as the aspect ratio tends to zero. Such an equation is obtained in a regime.
Numerical upscaling of problems with multiple scale structures have attracted increasing attention in recent years. In particular, problems with non-separable scales pose a great challenge to mathematical analysis and simulation. Most existing methods are either based on the assumption of scale separation or heuristic arguments.
In this talk, we present rigorous results on homogenization of partial differential equations with $L^{\infty}$ coefficients which allow for a continuum of spatial and temporal scales. We propose a new type of compensation phenomena for elliptic, parabolic, and hyperbolic equations. The main idea is: through the use of the so-called "harmonic coordinates" ("caloric coordinates" in the parabolic case), the solutions of the differential equations have one more degree of differentiability. It can be deduced from this compensation phenomenon that numerical homogenization methods formulated as oscillating finite elements can converge in the presence of a continuum of scales, if one uses global harmonic coordinates to obtain the test functions instead of using solutions of a local cell problem.
We investigate conserved models for phase transitions of Penrose-Fife type. Global well-posedness of strong solutions is obtained in the sense of $L_p$. Furthermore it can be shown that each solution converges to a steady state as time tends to infinty.
I will first discuss and present a proof of the following statement. Let $m>1$. Any $W^{2,m}$ isometric immersion from a domain $R^m$ into $R^{m+1}$ is $C1$ regular inside the domain. I will proceed to dicuss some open problems. In particular strategies and obstacles in generalizing this result to isometric immersions of any codimension will be discussed.
In nature one encounters patterns everywhere, from the stripes on zebras and the sand patterns in a desert, to the beating of your heart and the fingerprints on your fingers. A pattern forming system that has received extensive attention from experimentalists in recent decades is the diblock copolymer melt. Two types of polymer molecules are chemically bonded together to form a diblock copolymer. These molecules mutually repel each other, but due to the chemical bond they are restricted in their movement away from each other. These competing influences lead to pattern formation on a length scale between that of the system and that of the molecules. In recent years also mathematicians began studying models for diblock copolymer melts.
We are investigating a model for the simplest extension of this system, a blend of diblock copolymers and a third type of polymer, called homopolymer, which is not bonded to the copolymer molecules. This blend-model, which is variational in nature, is now well-understood in one dimension. We know how minimisers of the energy look and under which conditions they are non-unique. In higher dimensions we have bounds on the energy and we investigated the stability of some specific morphologies in two dimensions. Our study shows an interesting dependence of stability in these cases on the strength of the mutual repulsion between the different types of polymer molecules.
In a first part I will present a variational model for quasistatic crack growth in the presence of a cohesive force exerted between the lips of the crack. Next I will present a quasistatic evolutionary model based on a local stability criterion and obtained by viscous regularization. For both models I make the assumption that the crack path is prescribed.
This talk collects some results taken from my PhD Thesis and more in detail is related to a joint work with Gianni Dal Maso and a collaboration with Rodica Toader.
The time spent by a ferromagnetic system, ruled by a Kac potential $J_{\gamma}$, in one of the two phases has the order of magnitude of 1 divided by the probability of observing such a strong fluctuation that an interface is produced and move until the opposite phase is reached. That probability is exponentially small in the intensity $\gamma$ of the potential and the free energy gap between one of the phase and the next stationary state, where both phases are present in the same quantity separated by an interface.
Let $S\subset R2$ be a bounded Lipschitz domain and set $$ W^{2,2}_{iso}(S; R3) = \{u\in W^{2,2}(S; R3): (\nabla u)^T(\nabla u) = Id\ \text{ a.e.}\}. $$ Under an additional regularity condition on the boundary $\partial S$ (which is satisfied if it is piecewise continuously differentiable) we prove that the $W^{2,2}$ closure of $W^{2,2}_{iso}(S; R3)\cap C^{\infty}(\bar{S}; R3)$ agrees with $W^{2,2}_{iso}(S; R3)$.
I will present recent results about magnetic force formulae for two rigid magnetic bodies. Using different scales to describe the distance between the bodies, we discuss continuum limits of atomistic dipole-dipole interactions. This allows to link a classical force formula and a limiting force formula obtained earlier as a discrete-to-continuum limit for bodies being in contact on the atomistic scale.
This is joint work with Bernd Schmidt, cf. MPI MIS Preprint 29/2007.
In 2001 Koch and Tataru proved local (in time) existence of regular solutions to the Navier-Stokes equations for initial data in $vmo^{-1}$. In this talk I would present some uniqueness result of the solutions. I would also review related results if time permits.
In this talk we will investigate the regularity of the free boundary for a parabolic free boundary problem with constant gradient condition on the free boundary.
In particular, we will show that flat free boundary points are regular (in space). The proof is similar to, although somewhat simpler than, a proof used by Alt and Caffarelli to prove a similar result for elliptic problems.
This is a jount work with G.S. Weiss.
We consider the incompressible Navier-Stokes equations with initial data in amalgam spaces in which we can control the local singularities and decays of functions separately. We show the local existences of the solutions of the equations in those spaces.
We prove existence and uniqueness of local strong solutions for a model of capillary compressible fluids derived by J.E. Dunn and J. Serrin (1985). This nonlinear problem is approached by proving maximal regularity for a related linear problem in order to formulate a fixed point equation, which is solved by the contraction mapping principle. Localising the linear problem leads to model problems in full and half space, which are treated by Dore-Venni-Theory, real interpolation and ${\cal H}^\infty$-calculus. For these steps, it is decisive to find conditions on the inhomogeneities being necessary and sufficient.
We introduce some probabilistic concepts of Bose-Einstein condensation. In particular we discuss probability distributions on cycles. Here cycles do correspond to permutations of finitely many elements. As we give this week also a talk at the Berlin-Leipzig seminar, we focus in the first part on an introduction to Bose-Einstein condensation, it's meaning and mathematical justification. Aim is to acquaint the audience with the notion Bose-Einstein condensation.
We consider a two-dimensional model for a nonlinearly elastic plate where only a portion of the lateral face is subjected to boundary conditions of von Karman type, the remaining portion being free and we establish an existence theorem for these equations.
We consider the instationary Cahn-Hilliard equation, which describes the phase separation of a two-component alloys. In contrast to most mathematical treatments, which deal with a smooth free energy, we consider a logarithmic or similar singular free energy. Such a free energy naturally occurs in physics and has the mathematical property that it ensures that the concentration difference stays in the physical reasonable regime. In contrast to known proofs for existence of unique weak solutions, which are based on a truncation of the singularity in the free energy, we present a direct approach via Lipschitz perturbations of monotone operators. Based on these results, we prove that a solution of the instationary system converges to a solution of the stationary as time goes to infinity. Finally, we discuss how this method can be used to solve Navier-Stokes-Cahn-Hilliard type equations arising in phase field models for two-phase flows of viscous, incompressible, immiscible fluids.
The rigidity result of Friesecke-James-Mueller asserts that given any deformation in the Euclidean space of dimension n, the L2 distance of its gradient from a single rotation matrix is bounded by a multiple of the L2 distance from the group SO(n) of rotations.
We will present a program to adapt the result and the method of proof to the hyperbolic geometry. We will present the expected result in the hyperboloid model, which we will describe for our purpose. Some progress and technical difficulties will be discussed in detail.
We provide a Liouville type theorem in the framework of fracture mechanics, and more precisely in the theory of $SBV$ deformations for cracked bodies. We prove the following rigidity result: if $u$ is a $SBV$ deformation of $\Om$ whose associated crack $J_u$ has finite energy in the sense of Griffith's theory (i.e., $H^{N-1}(J_u)
A viscous shock wave is a traveling wave solution of a system of conservation laws, considered with (partial or total) parabolic regularization (coming, for example, from viscosity, heat conduction, etc). Navier-Stokes equations for a compressible flow or the equations of magnetohydrodynamics are two examples of such systems. In this talk the asymptotic behavior of a perturbation of such a shock will be discussed.
We obtain the interior regularity criteria for the vorticity of ``suitable'' weak solutions to the Navier-Stokes equations. We prove that if two components of a vorticiy belongs to $L^{q,p}_{t,x}$ in a neighborhood of an interior point with $3/p+2/q\leq 2$ and $3/2
We consider an ideal non-thermal elastic medium described by a stored-energy function. We study time-dependent configurations with subsonically moving phase boundaries across which, in addition to the classical jump relations of Rankine-Hugoniot type, some kinetic rule acts as a two-sided boundary condition. We establish a concise version of a normal-modes determinant that characterizes the local-in-time linear and nonlinear (in)stability of such patterns. Specific attention is given to the case where the enrgy function has two local minimizers which can coexist via a static planar phase boundary. Being dynamic perturbations of such configurations of particular interest, it is shown that the stability behaviour of corresponding almost-static phase boundaries is uniformly controlled by an explicit expression that can be determined from derivatives of the energy function and the kinetic rule at the minimizers.
In this talk I will present results about energy minimizers in different regimes of thickness. Moreover I will show a theorem on the $\Gamma$-limit as the diameter tends to zero.
The Bellman equation arises in stochastic control theory when one is looking for the best possible choice of a control function which minimises a cost functional associated to a stochastic process. On the analytic side this is described by a fully non-linear elliptic equation. When the control of Wiener processes is considered, usual second order elliptic differential operators occur in the equation. But considering Markov jump processes non-local integro-differential operators enter into the equation. These operators are similar to fractional powers of the Laplacian and obey a maximum principle, which allows to prove existence of positive solutions of the corresponding Bellman equation.
We study the problem of characterizing quasiconvex hulls for three solenoidal (divergence free) wells in dimension three when the wells are pairwise incompatible. We provide the characterization for a generic regime by translating the problem into the language of H-measures. As a by-product we obtain the characterization of a class of three-point H-measures associated with three phase mixtures of characteristic functions in dimension three.
(This is a joint work with V. P. Smyshlyaev).
An alternate to the Cahn-Hillard model of phase seperation for two-phase systems in a simplified isothermal case in given. It introduces nonlocal terms and allows reasonable bounds for the concentrations. Using the free energie as Lyapunov functional the asymptotic state of the system is investigated and characterized by a variational principle.
The present paradigm of magnetic data storage is approaching its fundamental limits for areal storage density, as well as for speed in data processing. As a result, there is an urgent need for reliable alternatives to current magnetic recording media, which are based on longitudinal thin films, and to conventional mechanism of magnetization reversals, based on damping switchings. In this talk, faster modes of magnetization reversals, using precessional magnetization motion, are discussed in the context of traditional longitudinal media and its promising alternatives: perpendicular and patterned media. This analysis uses Landau-Lifshitz type equations to describe the magnetization dynamics.
Wigner/Semiclassical measures are a standard tool for the study of propagation of concentration and oscillation effects for solutions to wave equations. We shall illustrate this theory adressing two specific issues:
1. Discrete Wave equations on the torus. We shall give a discrete version of the theory of Wigner measures and apply it to characterize the high frequency behavior of discrete waves, recovering in particular classical results on group velocity.
2. Characterization of Semiclassical measures associated to solutions of the Schr"odinger equation on compact manifolds. In particular, we shall focus on the relation between the dynamics of the geodesic flow on a compact manifold and the structure of Wigner measures associated to the corresponding Schrödinger equation.
A two-phase segregation model in solids under isothermal conditions is derived where due to plastic effects the number of vacancies changes when crossing a transition layer. The thermodynamical correctness of the model is shown and the existence of weak solutions in suitable spaces is reviewed. By a formal asymptotic analysis the dynamics of the interface and its dependence on the unsymmetric vacancy distribution is studied.
As minimal surfaces, free boundaries are not always regular. Therefore the regularity theory for free boundaries naturally splits into two branches, finding criterias that implies regularity and classification of possible singular points. This talk will focus on the second branch for a problem considered by R. Monneau and G.S. Weiss. They consider the problem \begin{displaymath} \Delta u = -\chi_{\}u>0\}}, \end{displaymath} $\chi_{\{u>0\}}$ is the characteristic function of the indicated set. They show that the free boundary is regular whenever the solution grows quadraticly away from the free boundary. This property is shown for some special kinds of solutions, for instance for variational solutions. However they leave the question open wether this quadratic growth is true in general.In this talk we will recapitualte the main results of Monneau's and Weiss' paper. We will also answer their open question and show the existence of non-regular free boundary points. The proof is based on Schauders' fixed point theorem and symetry. The existence of both kinds of non-regular free boundary points will be shown. The first kind is when the solution grows faster than quadraticly away from the free boundary, at these points the solution fails to have the optimal $C^{1,1}$-regularity and will be denoted singular points of the solution. The other kind is the degenerete points when the solution grows slower than quadratic. This work is joint with G.S. Weiss
We provide a partial generalisation of the classic theorem of Marstrand that states, given a Radon measure in Euclidean space with positive finite s-density and almost all point s, s has to be an integer. Our theorem holds for all finite dimensional normed vector spaces whose unit ball is a polytope, over the restricted density range s in [0,2].
Moser Trudinger inequality comes in the limit case of Sobolev embedding.My talk will be on the derivation of an improved M-T inequality and then its application to prove the existence of principal eigenvalues for semilinear and quasilinear elliptic indefinite weight eigen value problem in the limiting dimension of sobolev imbeddings.Some more extension to higher order elliptic problem will also be discussed.
We consider the flow of two immiscible, incompressible viscous fluids of same density but different viscosities. There are many results on short-time existence of (strong) solutions for this and similar free boundary value problems in fluid mechanics. Also the case of global in time existence of strong solutions for initial data near an equilibrium was studied by many mathematicians. But there are only few results on large time existence of (weak) solutions to such kind of free boundary value problems for arbitrary initial data.
In this talk we will discuss some results on large time existence of suitably defined weak solutions to the two phase flow described above. Neglecting the effect of surface tension, weak solutions are known to exist for arbitrary large data and time; but there is almost no information on the interface between the two fluids in this case. Taking surface tension into account gives an a priori bound of the area of the interface, which can be used in the construction of weak solutions, which have a countably rectifiable interface of finite area. We present a conditional result showing the existence of weak solutions in the case of surface tension if no area or energy is lost during the approximation.
We present a theory of gradient flows in asymmetric metric spaces presupposing no linear or differential structure. This extends the classical theory in inner product spaces and the theory in metric spaces recently proposed by Ambrosio, Gigli and Savare. We begin by extending to the asymmetric situation notions that arise in analysis on metric spaces (such as absolute continuity, metric derivaties, local slopes and curves of maximal slope). In particular, curves of maximal slope are a natural extension of gradient flows to asymmetric spaces. Then, we study variational approximations (time discretizations, error estimates and convergence of solutions of the discrete problem) of curves of maximal slope. Finally, we specialize our results to the context of asymmetric extensions of Wasserstein spaces of probability measures. We end by outlining possible applications to solid mechanics. The motivating observation is that the time evolution of probability measures that arise in the study of materials with microstructure, phase transformations, plasticity or damage is better described on asymmetrically metrizable (but not metrizable) topological spaces than on metrizable topological spaces.
This is joint work with Marc Oliver Rieger, University of Zurich and Johannes Zimmer, University of Bath.
The first microscopic theory of superfluidity was originally found in 1947 by Bogoliubov in three revolutionary papers on the theory of interacting Bose gas. His Weakly Imperfect Bose Gas (WIBG) coming from the truncation of a full interacting Bose gas was a starting point for this theory. However, only very few rigorous results concerning his WIBG and ansatzs were previously done until 1998-2000 where a first rigorous analysis of this Bogoliubov model (WIBG) was found at all temperatures and densities. The aim of this talk is to do a more deep analysis of the Bogoliubov theory, including all recent studies (2001) and some new criticizes (2002-2004). Actually, this more detailed analysis gives rise to a new microscopic theory of superfluidity at all temperatures (2004) then introduced at the end of this talk. In particular, the talk should be concluded by the corresponding phase diagram of this new theory: it exhibits the ''Landau-type'' excitation spectrum in the presence of a depleted Bose condensation for small temperatures with the formation of "Cooper-type pairs", even at zero-temperature (experimentally, an estimate of the fraction of condensate in liquid $^{4}$He at T=0 K is $9$ $\%$).
We start with discussing a free boundary value problem, which describes the motion of a infinite ocean of water bounded by a fixed bottom below and a free surface above. Using the contraction mapping principle the existence of (strong) solutions locally in time can be proved once the corresponding linearized system can be solved in suitable $L^p$-Sobolev spaces in certain class of unbounded domains. In the main part of the talk we present a method to solve this linearized system - the generalized instationary Stokes system - by means of semi-group theory. This leads to the question whether the associated Stokes operator has so-called maximal $L^p$-regularity. By a famous result due Dore and Venni it is sufficient to prove the existence of bounded imaginary powers of the Stokes operator, which can be done by pseudodifferential operator techniques.
The elastic energy of a thin elastic sheet is the sum of a non-convex term that penalizes local stretching (and compression) and of a small (depending on the thickness) singular perturbation that takes bending into account.
When the sheet is confined in a region of small diameter, these two terms enter in competition and a concentration of curvature on lines and points is observed.
By using a combination of explicit constructions and general results from differential geometry we obtain an upper bound for the elastic energy of a compressed sheet in terms of its thickness at the power $5/3$. We show that this exponent is optimal in certain simplified geometries, that are conjectured to represent the canonical singularities leading to this exotic power law.
Consider the Cauchy problem for the two following soliton equations: Korteweg-de-Vries with a small dispersion parameter epsilon and the nonlinear Schroedinger equation in 1+1 dimensions with cubic nonlinearity in the so-called semiclassical limit. It is known that at a certain "caustic" there is a "phase transition" in the solution of each of these equations. Oscillations of high frequency appear, so strong limits no longer exist. Soliton theory enables us to provide detailed information for the solutions as the small parameter epsilon goes to zero, prove the existence of weak limits, and describe completely the different oscillatory regions in terms of a variational problem of "electrostatic type". In the case of the focusing NLS this is a non-convex problem.
We introduce two probabilistic models for $N$ interacting Brownian motions moving in a trap in $ \R^d $ under the presence of mutually repellent forces. The two models are defined in terms of transformed path measures on finite time intervals under a trap Hamiltonian and two respective pair-interaction Hamiltonians. The first pair interaction exhibits a particle repellence, while the second one imposes a path repellency. We analyse both models in the limit of diverging time with fixed number $ N $ of Brownian motions. In particular, we prove large deviations principles for the normalised occupation measures. The minimisers of the rate functions in the two cases are given in terms of the ground state, respectively the ground product-states, of a certain associated operator, the Hamilton operator for a system of $ N $ interacting trapped particles (bosons). In the case of path-repellency, we also discuss the case of a Dirac-type interaction, which is rigorously defined in terms of Brownian intersection local times. We prove a large-deviation result for a discrete toy model. This study is a contribution to the search for a mathematical formulation of the quantum system of $ N $ trapped interacting bosons as a model for Bose-Einstein condensation, motivated by the success of the famous 1995 experiments. Recently, Lieb et al. described the large-$N$ behaviour of the unrestricted ground states in terms of the well-known Gross-Pitaevskii formula, involving the scattering length of the pair potential. We prove that the large-$N$ behaviour of the product-state ground states is also described by the Gross-Pitaevskii formula, however with the scattering length of the pair potential replaced by its integral.
The regularity results for diffusion operators with irregular coefficients (mainly by De Giorgi, Fabes, Krylov, Moser, Nash, Safonov, Stroock) have had a great and wide impact in Analysis. In the talk we report on successful attempts to build a similar theory for jump processes.
The study of effective interface models have started with continuous Ising models. The interface is modelled as a random height function representing the distance of the interface from a reference height. The interaction depends only on the gradient, hence due to the rich symmetry phase transitions occur. For the latter one needs that the interaction is strictly convex, because this ensure the existence of infinite Gibbs measures at least for dimension greater equal than three. If one consider the so-called random field of gradients (of the height functions) they exist for any dimension.
In the talk we present a new approach for the random field of gradients and a large deviation result for the free boundary case. We show the existence of the specific entropy and free enrgy. We further discuss various problems connected with boundary conditions and outline the motivation for studying non convex interactions.
The talk is based on work with Deuschel and Sheffield.
We will we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller. We then give some applications to the problem of getting a quantitive estimate of how much convex integration solutions oscillate.
In this talk, I will present a neat application of computational algebra which has applications in the field of computer vision. The main question we will answer is: "Is a configurations of point uniquely determined (up to rigid motion) by the distribution of the pairwise distances between its points." This work is in collaboration with Gregor Kemper (TU Muenchen).
There are considered the Navier-Stokes equations (NSE) with a zero (or potential) external force vector. The group of transformations admitted by these equations is infinite-dimensional: coefficients of operators of the corresponding Lie algebra contain four arbitrary functions of time. Rich properties of the NSE symmetry produce a lot of their exact solutions. Classical examples of invariant solutions of the Navier-Stokes equations are the solutions by Poiseuille, Couette and Hamel-Jeffery. Notion of a partially invariant solution of a system of differential equations, introduced by Ovsiannikov, allowed us to extend a class of exact solutions of the NSE essentially. However this class is not exhausted by invariant and partially invariant solutions. Its further extension could be obtained with a help of a two-step procedure: first we reduce the initial NSE model to a partially invariant sub-model of less dimension, which inherits a part of the group, admitted by the NSE, and then we construct invariant solutions of the reduced sub-model. Following this procedure, there was revealed the group-theoretical nature of the famous von Karman solution.
Symmetry approach turned to be especially useful in application to the free boundary problems for the NSE. Local theory of specified problems is close to be completed nowadays. At the same time, the global solvability of non-stationary problems in time is proved only under the condition that the initial velocity vector is small. Solvability of stationary problems is established under restrictions of smallness of Reynolds number or capillary number. Employment of properties of invariance of free boundary conditions allowed us to obtain non-local results in problems with a free boundary, which effective dimension does not exceed 2. Non-stationary problems on a rotating ring and on the viscous layer spread over a rotating plane as well as analogues of stationary flows of Hamel-Jeffery and Poiseuille are among this kind of problems. Moreover, with a help of partially invariant solutions there are constructed examples of blowing up and shrinking of the flow domain phenomena in free boundary problems.
One more method to obtain new exact solutions of the NSE is the "raising" of the known solutions by reduction of their properties of invariance. As an example, there is considered a non-stationary analogue of the classical Hamel-Jeffery flow, which describes stationary self-similar flow in a flat diffusor. If we depart from the property of stationarity, but keep the self-similarity, we'll come to a non-standard boundary-valued problem in a stripe for a quasi-linear elliptical equation of fourth order, which is satisfied by a stream function of flow in self-similar variables. Its specificity consists in the fact that on the "right" infinity limit values of a stream function in dependence on the polar angle could be set arbitrarily, meanwhile on the "left" infinity they are determined by preserved in time value of flux of liquid through the cross-section of sector as a solution of Hamel-Jeffery problem. Case of zero flux represents a special interest for investigation. In this case the velocity field of flow possess a finite Dirichlet integral at any arbitrary positive time value.
Some "patterns" universally appear in any large measurable set in R^n, and the others can be avoided by such sets. (By "large" we mean a set of positive measure, or Borel set of sufficiently large Hausdorff dimension, etc.) Trivial examples of such results are that any set of positive measure realizes or sufficiently small distances by pairs of points in the set, or that any such set contains many copies geometrically similar to any prescribed finite set. On the other hand, even in R, a set of positive measure can avoid to contain a homothetic image of the sequence {1/n: n positive integer}.
For a particular "pattern" (and the notion of "large") to recognize if it is avoidable, or if it universally appears in each large measurable set, is typically very difficult question. Some new results and many open questions are discussed.
We study variational problems with level set constraints of the form
where and .
In the one-dimensional case sharp conditions for the existence of global
and local minimizers are derived. Moreover some existence results are
provided when the energy E(u) is not of integral form, but instead
satisifes some abstract conditions like additivity, translation invariance
and solvability of a transition problem.
In the general n-dimensional case we consider the -limit when
. The result turns out to be a nonlocal functional
with minimizers satisfying (in the isotropic case) the minimal interface
criterion.
We present a geometric singular perturbation analysis of a chemical oscillator. Although the studied model is rather simple, interesting dynamical phenomena like relaxation oscillations, canard solutions, and mixed mode oscillations occur. The problem is interesting in itself, however, the main theme of the talk is to explain recent developments in the dynamical systems approach to singular perturbation problems in the context of a specific example.
Smoluchowski's equations provide a mean-field description of several mass aggregation processes in nature such as the formation of clouds, stars and polymers. They also have mathematical applications to fields such as random graphs and population genetics. Nevertheless, little is known analytically about the existence of self-similar solutions.
In this talk, I will describe extremely simple proofs of self-similar blowup (gelation) and approach to equilibrium in one of the few solvable cases (additive and multiplicative kernels). The results have an appealing probabilistic interpretation. This is joint work with Bob Pego (Maryland).
I will talk about the steady Navier-Stokes equations in a two dimensional channel with slip boundary conditions. Under some restrictions on the shape of the domain and the friction I show existence of solutions with no bounds on largeness of the flux of the flow and the norm of the external force. The crucial point in the proof is the maximum principle for a reformulation of the problem. This technique gives a new type of estimates in the Hölder space for the solutions with the infinite Dirichlet integral.
We consider the flow of a hypersurface driven by its Gauß curvature. This geometric problem yields a parabolic PDE, a Monge-Ampère equation. We study this equation subject to the second boundary value condition.
In our talk, we will give an introduction to such equations, discuss the obliqueness of the boundary condition, shorttime existence, and a priori estimates required to prove longtime existence. We will study the longtime behavior of these solutions and prove -- depending on the setting -- either convergence to a hypersurface of prescribed Gauß curvature or to a translating solution. For these steps we use only PDE techniques.
If there will be some time left, we will report on results for similar flow equations and boundary conditions.
Part of this work is joint work with Knut Smoczyk.
We consider smooth, embedded, minimal n-hypersurfaces Mj converging to a stationary varifold Σ. Under the assumption that the second fundamental AMj of Mj is bounded in L2, that is we prove that the Hausdorff dimension of the singular set is estimated by
Intracellular Ca2+ patterns are spatio-temporal concentration profiles. Living cells exhibit transitions from localized structures (called sparks or puffs) to traveling waves. Waves may form rotating spirals or target patterns.
The Ca2+ concentration changes by nonlinear release and uptake of cell organelles. Release is accomplished through channels with stochastic transitions between the states closed, open and inhibited. The transition rates depend on Ca2+. Channels are coupled by Ca2+ diffusion and spatially organized in clusters.
In most cells and for most channel types the dynamics of the spatial Ca2+ profile is faster than the channel dynamics because flux densities are large and diffusion time for distances of channel spacing is small. That allows to eliminate the reaction-diffusion equation for Ca2+ adiabatically. That reduces the model to an array of stochastic elements with state dependent transition rates and coupling over a distance of a few cluster spacings. We give simple rules for calculating the transition rates.
Our approach provides a generalized stochastic model to which many biologically detailed models for the different Ca2+ channel types can be reduced as long as they fulfill the requirement of time scale separation.
In the limit of large numbers of channels per channel cluster and strong spatial coupling, continuum models can be applied. With such a continuum model, we investigate the impact of increased mitochondrial Ca2+ cycling on pattern formation and stability. Experimental findings can be explained by a gap in the dispersion relation for wave trains.
