Seminar on partial differential equations

The talk will deal with some models of phase segregation, based on two balance equations for, respectively, microforces and microenergy, and on fairly nonstandard but thermodynamically compatible constitutive choices. The first model is akin to the classical Allen-Cahn equation for an order parameter whose space-time evolution may lead to phase segregation by atomic rearrangement in the absence of diffusion; the second model generalizes the Cahn-Hilliard equation, that takes diffusion into account. The analysis of the related systems of two (partial) differential equations has been made the subject of an ongoing joint research program with G. Gilardi, P. Podio-Guidugli and J. Sprekels. The two unknowns entering the equations are the order parameter and the chemical potential. Existence, uniqueness and stability results will be discussed in my presentation.

Karel Vácha will present a report on the paper A.V. Kazhikhov, V.A. Weigant: On the global existence theorem for two-dimensional compressible viscous flows.

The existence of at least three solutions is obtained for a class of systems of elliptic equations involving the p-Laplacian with homogeneous Dirichlet boundary conditions. The results provide sign information for the solutions. The approach relies on the method of sub-supersolution and variational techniques.

The Griffith criterion is an energetic fracture criterion which is frequently applied to decide whether a preexisting crack in an elastic body is stationary for given external forces. In this lecture we model the evolution of a single crack in elastic materials as a rate-independent process based on the Griffith criterion. Due to the nonconvexity of the stored energy (with respect to the crack length), the crack evolution can be discontinuous in time and suitable jump criteria are needed. Via a vanishing viscosity approach we derive such criteria. Furthermore, we discuss the convergence of fully discretized viscous models (i.e. with respect to time and space) to the vanishing viscosity model. The goal is to derive a relation between the discretization parameters (i.e. the mesh size, the time step size, the viscosity parameter and the minimal crack increment) which guarantee the convergence of the scheme. The convergence proof relies on regularity estimates for the elastic fields close to the crack tip taking into account contact conditions on the crack faces and on local and global finite element error estimates.

It has been possible in the recent years to generalize parts of the linear Calderon-Zymund theory to the non-linear setting of the p-Laplacian. We first discuss the basic principles and results of this method. After that we show how to generalize this to the context of variable exponents.

The talk will deal with some models for ionized fluid mixtures. In the first part I will present the GENERIC thermodynamical framework and show some rather standard models in view of the framework. I will also discuss the relation between models using the generic framework and other models for fluid mixtures. In the second part I will present a proof of existence of a solution for a model for incompressible ionized fluid mixtures. I will put the main emphasis on apriori estimates.

We consider a system of elliptic and parabolic diagonal equations describing mostly the stochastic differential games. Such system however contains the term with quadratic growth in gradient of unknown on the right hand side. We present a simple method how the existence of a weak solution can be established in arbitrary dimension, provided the the right hand side satisfies certain structure assumptions that however cover the most class of stochastic differential games. Moreover, since the method is elementary, we can extend the procedure to consider a nonlinear systems (nonlinear in the leading part) of the type p-Laplacian with p-growth on the right hand side and for similar strucuture asumptions we can prove the existence of a weak solution even for such nonlinear systems.

We consider the nonlinear diffusion equation $u_t=u^p u_{xx}$ for $p>0$. We construct positive classical solutions which are of the form \[ u(x,t) = (-t)^{-\frac{1}{p}} F(x+\frac{1}{p\alpha}\ln (-t)), \qquad x\in\R, \ t<0, \] with arbitrary $\alpha>0$, by solving an associated ODE for $F$. These `ancient slowly traveling wave solutions' have the following properties:
1.) \ If $p\le 2$ then $u$ blows up at time $t=0$ with empty blow-up set.
2.) \ If $p>2$ then $u$ can be extended so as to become an entire positive classical solution $\bar{u}$, defined on $\R\times \R$, such that $\bar{u}_x >0$ on $\R$ and \[ \bar{u}(x,t) \to 0 \qquad \mbox{as } t\to\pm \infty, \] locally uniformly with respect to $x\in\R$.
These 'homoclinic orbits' are unbounded in space at each time; this inconvenience cannot be removed, because a result of Liouville type asserts that any positive entire solution of $u_t=u^p u_{xx}$ must be a constant whenever $p\ge 0$.

In this talk we will give an asymptotic description of the profile of standing waves (traveling wave solutions with zero wave speed) of lattice differential equations (an infinite system of ordinary differential equations). The proof will be based on new results about the asymptotic behavior of the solutions of asymptotically autonomous linear difference equations with continuous time.

In the talk I will consider suitable weak solutions to the 3D Navier-Stokes equations. For such solutions the size of a putative singular set can be limited in terms of both Hausdorff and the box-counting dimensions. I will sketch main ideas behind such restrictions on the size of a singular set and then study their consequences for the uniqueness of Lagrangian trajectories corresponding to a given weak solution.

In this talk, we present a new model describing phase separation and damage phenomena. We introduce a suitable notion of weak solutions and provide existence results for the introduced model. This is a joint work with C. Heinemann (WIAS, Berlin).

A classical result of Caffarelli, Kohn, and Nirenberg states that the one dimensional Hausdorff measure of singularities of a suitable weak solution of the Navier-Stokes system is zero. We present a short proof of the partial regularity result which allows the force to belong to a lower regularity Lebesgue space.

A survey will be given of recent developments in the theory of generalised trigonometric functions. These functions play an important role in questions involving the p-Laplacian and also have considerable intrinsic interest: several outstanding questions remain unresolved and will be discussed. Other approaches to the Dirichlet problem for the p-Laplacian will also be considered.

A survey will be given of recent developments in the theory of generalised trigonometric functions. These functions play an important role in questions involving the p-Laplacian and also have considerable intrinsic interest: several outstanding questions remain unresolved and will be discussed. Other approaches to the Dirichlet problem for the p-Laplacian will also be considered.

In this talk, recent results concerning the global regularity of initial-boundary value problem to some 2D coupled Navier-Stokes equations are presented, with emphasis on the variable viscocity coefficients.

The mechanics of a thin elastic sheet can be explored variationally, by minimizing the sum of "membrane" and "bending" energy. For some loading conditions, the minimizer develops increasingly fine-scale wrinkles as the sheet thickness tends to 0. While the optimal wrinkle pattern is probably available only numerically, the qualitative features of the pattern can be explored by examining how the minimum energy scales with the sheet thickness. I will introduce this viewpoint by discussing previous results by Kohn and others on simpler but related problems. Then I'll discuss recent work with Robert Kohn, concerning the wrinkles observed by Davidovitch et al in radially loaded annular elastic thin film (arxiv, 2010).

We describe results and problems for function spaces of Besov-Sobolev type on bounded smooth and non-smooth domains in the Euclidean n-space. We concentrate on the following topics:

• 1. Definitions by restriction and intrinsic
• 2. The extension problem
• 3. Traces on the boundary
• 4. Distinguished subspaces, duality
• 5. Compact embeddings