Seminar on partial differential equations

We study when the composition $u\circ f$ of Sobolev function $u$ and homeomorphism $f$ is in some suitable Sobolev space. We show that if $f$ has finite distortion and $q$-distortion $K_q=|Df|^q/J_f$ is integrable enough, then the composition operator $T_f(u)=u\circ f$ maps functions from $W^{1,q}_{loc}$ into space $W^{1,p}_{loc}$. To prove it we need to know when the inverse mapping $f^{-1}$ maps sets of measure zero onto sets of measure zero (satisfies Luzin $(N^{-1})$ condition).

We show optimal results concerning the integrability of gradients of weak solutions to elliptic problem of Stokes type with respect to the integrability of the right hand side function. This is joint work with Lars Diening from University of Freiburg.

organized by Šárka Nečasová, Milan Pokorný, Tomáš Roubíček

We treat the Navier-Stokes equations in a general unbounded domain. Such domains do not satisfy any geometric condition except for smoothness of the boundary, so in particular we consider domains which are not necessarily exterior or aperture domains or perturbed half spaces. For those general domains we develop a theory of very weak solutions, which is well established in the case of bounded domains, where as solution class one has $L^r(0,T;L^q(\Omega))$ with Serrin exponents $r$ and $q$. For unbounded domains, the $L^q$-Stokes operator is not defined in a general unbounded domain, so we will make use of an approach by Farwig, Kozono, Sohr, replacing the space $L^q(\Omega)$ by $\widetilde{L}^q(\Omega)$ of functions which behave locally like $L^q$-functions but globally like $L^2$-functions. In the talk, local existence and global uniqueness of very weak solutions in the class $L^r(0,T;\widetilde{L}^q(\Omega))$ is demonstrated, where $r$ and $q$ are Serrin exponents, i.e. $2/r+n/q=1$, $2<r<\infty$, $n<q<\infty$.

Many processes in nature can be considered as rate-independent. A wide class of such processes can be described by balance equations containing only two quantities that need to be chosen constitutivly, namely the Gibbs free energy and the dissipation potential. For that class of processes a advantegeous formulation, namely the energetic formulation has been proposes by Mielke et al. This formulation is advategeous not only from the point of view of analysis but also from numerical point od view as it allows for a convenient time-discretization. Solving the time-discretized problem numerically however implies searching for global optima a often non-convex, non-smooth functional. In this talk we compare several global optimization strategies and propose an improvement of these suitable for the class of problems considered by ensuring the energy balance to hold. Numerical examples will also be given.

Given a finite sequence of times $0<t_1<...<t_N$, we construct an certain example of a smooth solution of the Boussinesq equations which describe heat convection in a viscous incompressible fluid under the influence of gravity in $\mathbb{R}^d$, $d=2,3$. This solution is spatially poorly localized at the beginning of the evolution but tends to concentrate until, as the time $t$ approaches $t_1$, it becomes well-localized. Then the solution spreads out again after $t_1$, and such concentration-diffusion phenomena are later reproduced near the instants $t_2,t_3,...,t_N$.

A model for incompressible ionized mixtures of constituents with different molar weights is presented. The derivation of the model and formal assumptions will be presented. Finally a proof of existence of a weak solution to the problem will be presented.

We present some recent progress on the one-dimensional motions of compressible fluid coupled to radiation through the radiative transfert equation. Global existence results are presented and difficulties concerning the large-time behavior are evoqued.

We will describe the main ingredients of a variational approach to the Navier-Stokes system that consists in setting up an error functional measuring the departure of fields from being a solution. The strategy to show the existence of (regular) weak solutions consists of:

1. Show that there are global minimizers for the error in suitable clases of fields. This involves coercivity and convexity. There is no problem with the second, but the first is crucial, and indeed is the single, most relevant difference between dimension 2 and 3.
2. By using optimality, one can try to show that critical points of the error functional can only occur at zero error. Proving this leads to care about the linearized system.

This strategy has already been used successfully in other circumstances like ODEs, and controllability problems. The whole point of this seminar is to explore to what extent this same approach may be helpful for the Navier-Stokes system.

We consider Bellman systems to stochastic di erential games with respect to Stackelberg equilibriums. In this setting, in contrast to Nash equilibrium, there is a hierarchical order of players where certain players are aware of the strategy of certain other competitors. This leads to Bellman systems

where the right hand side may grow quadratically and carries new difficulties compared to Nash games since the known regularity theory for elliptic nonlinear systems is not applicable.

The systems in consideration satisfy certain positivity conditions for the Hamiltonians H_ν. They are sufficient to guarantee the existence of C^α-regularity and W^2,p-solutions.

Many complex fluids have a yield stress, i.e. a critical stress which needs to be applied before the fluids will flow. The concept of "the" yield stress, however, is often an oversimplification: Observed values of the yield stress often depend on the duration for which stress is applied or on prior flow history. This has led to the claim that the yield stress is a "myth" and yield stress behavior should arise only as a limiting case of a more "fundamental" theory.

In this talk, I shall formulate and analyze a model which can describe fluids with a nonmonotone shear stress / shear rate relationship. It is shown how yield stress fluids arise as a limiting case of such a model. Differences between critical stresses for "fast" and "slow" yielding, hysteresis phenomena and thixotropy are naturally explained by the model.

We discuss three different approaches to the generalisation of doubly nonlinear inclusions to functions of bounded variation. These are the energetic formulation introduced by A.Mielke and F.Theil (2004), a subdifferential formulation recently examined by U.Stefanelli (2009) and a formulation based on the Kurzweil integral due to P.Krejčí and M.Liero (2009). We will show similarities and differences between these approaches. Especially we are going to consider their behaviour at jump points, the behaviour in the case of convex and non-convex energies and illustrate this by a number of examples.

Joint work with M. Day and J Qing

Based on the construction of Bourgain and Pavlovic, I will show that the solutions to the Cauchy problem for the three dimensional different types of norm inflations in ${\dot B}_{\infty}^{-1, \infty}$. Particularly the magnetic field can develop norm inflation in short time even when the velocity remains small and vice verse.

Joint work with L. Brandolese

I will analyze the decay of weak and strong solutions to the three-dimensional viscous Boussinesq system. I will show that generic solutions blow-up as $t\to\infty$ in the sense that the energy and the $L^p$-norms of the velocity field grow to infinity for large time, for $1\le p<3$.

In the case of strong solutions sharp estimates can be provided both from above and from below and explicit asymptotic profiles can be found.

Finally I will consider solutions arising from $(u_0,\theta_0)$ with zero-mean for the initial temperature $\theta_0$ and show that they have a special behavior as $|x|$ or $t$ tends to infinity: contrarily to the generic case, their energy dissipates to zero for large time.

I would like to study the total variation flow and other equations with a so-called sudden directional diffusion in the simplest mono-dimensional case. These systems arise from the theory of crystal growth and image processing. Our goal is to introduce a new approach, which will redefine the basic quantities in mathematical analysis from the point of view of PDEs. The main idea is to treat the special solutions which in the classical framework are singular as the most smooth. This point of view gives us some hits to build our new approach. The talk will base on joint results with Piotr Rybka (Uni of Warsaw).

It is known that by supplementing a reaction-diffusions system (with classical mixed Dirichlet-Neumann boundary conditions) on some part of the boundary by so-called Signorini boundary conditions, new bifurcation points of stationary solutions arise, even in a parameter domain where the trivial solution of the classical problem is stable.

The talk is a survey on results which state that, roughly speaking, one also has such a bifurcation, even in a global sense, if instead of Signorini conditions one uses inclusions which can describe an obstacle or regulation more realistically.

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modelled by the incompressible Navier-Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. Here we prove existence of stationary solutions for this problem for the simplified situation where the body is replaced by a source term of compact support, and then based on the existence result, the asymptotic behavior is also analyzed.

Nematic elastomers are systems which combine optical properties of nematic liquid crystals with the mechanical properties of rubbery solids. They display phase transformations, material instabilities, and microstructures. These phenomena are related to the formation of elastic shear bands which are reminiscent of mechanical twinning in shape-memory alloys. The richness of the underlying material symmetries makes the mathematical analysis of this system particularly rewarding.

In this talk, we will review the recent progress on the modelling of martensitic-like microstructures in nematic elastomers, which has led to accurate coarse-grained models for the effective mechanical response.

The goal of this work is to obtain homogenized models for elastic media with fluid inclusions. First a model where the gaseous bubbles are not connected to each other is presented, and next the case when the fluid inclusions are connected through a diadic tree is investigated in 1D and 3D.

Consider a class of reaction-diffusion systems of activator-inhibitor type which is subject to Turing's diffusion-driven instability. It is known that unilateral obstacle lead to bifurcation of stationary spatially nonhomogeneous solutions in a parameter domain where the trivial solution of the unperturbed problem is stable. It was for many years an open question whether an analogue result holds in the absence of any Dirichlet conditions. In the talk the (perhaps surprising) answer to this question is given.

The talk concerns local (nearby of non-degenerate solutions) transformations of Signorini problems into mixed boundary value problems for semilinear elliptic PDEs. The mixed boundary value problems can be written as smooth operator equations, to which the classical Newton iteration procedure can be applied.

The challenge of describing the complexity of nematic liquid crystals through a model that is both comprehensive and simple enough to manipulate efficiently has led to the existence of several major competing theories. One of the most popular (among physicists) theory was proposed by Pierre Gilles de Gennes and was a major reason for awarding him a Nobel prize in 1991. The theory models liquid crystals as functions defined on a two or three dimensional domains with values in the space of Q-tensors (that is symmetric, traceless, three-by-three matrices). Despite its popularity with physicists the theory has received little attention from mathematicians until a few years ago when John Ball initiated its study. Nowadays it is a fast developing area and these lecture will survey its development.

This is the first lecture of the short course, see the complete schedule.

We introduce the notion of cocompactness, a property which is weaker than but related to compactness. Differential equations give rise to non-compact problems when the equation possesses a non-compact invariance, such as translation invariance or a scale invariance. In presence of such transformations, sequences converge only modulo countably many "escaping-by-invariance" terms, giving rise, in particular, to concentration phenomena. We survey several applications to semilinear elliptic problems and dispersive equations.

In this lecture we discuss the existence, stability and asymptotic profile at space infinity of a viscous flow around a rigid body in 3D. The case where the body is translating has been intensely studied, while less results have been established when the body is rotating though it is a physically relevant situation. We want to get better understanding of effect due to rotation from qualitative properties of the flow. There are two remarkable aspects. One is a certain hyperbolic feature that is observed in the spectrum of generator, and the other is an important role of the axis of rotation along which the leading term of the flow is largely concentrated. We address the 2D problem as well. The exterior steady problem in 2D is quite difficult because of the Stokes paradox when the body is at rest. It is made clear that the situation becomes less difficult when the body is rotating, and this is related to the latter aspect mentioned above.

In this lecture we discuss the existence, stability and asymptotic profile at space infinity of a viscous flow around a rigid body in 3D. The case where the body is translating has been intensely studied, while less results have been established when the body is rotating though it is a physically relevant situation. We want to get better understanding of effect due to rotation from qualitative properties of the flow. There are two remarkable aspects. One is a certain hyperbolic feature that is observed in the spectrum of generator, and the other is an important role of the axis of rotation along which the leading term of the flow is largely concentrated. We address the 2D problem as well. The exterior steady problem in 2D is quite difficult because of the Stokes paradox when the body is at rest. It is made clear that the situation becomes less difficult when the body is rotating, and this is related to the latter aspect mentioned above.

A parabolic obstacle-type problem with no restriction on the sign of the solution is considered. More precisely, we study the regularity properties of a solution and of the free boundary in a neighborhood of the fixed boundary of a domain. The exact mathematical formulation is as follows:

Let function $u$ and an open set $\Omega \subset \mathbb{R}^{n+1}_+$ solve the problem $$ \begin{aligned} H[u]=\chi_{\Omega} \quad &\text{in}\quad Q_1^+,\\ u=|Du|=0 \quad &\text{in}\quad Q_1^+ \setminus \Omega,\qquad\qquad (*)\\ u=0 \quad &\text{on}\quad \left\lbrace x_1=0\right\rbrace \cap Q_1, \end{aligned} $$ where $H=\Delta -\partial_t$ is the heat operator, $\chi_{\Omega}$ denotes the characteristic function of $\Omega$, $Q_1$ is the unit cylinder in $\mathbb{R}^{n+1}$, $Q_1^+=Q_1 \cap \left\lbrace x_1>0\right\rbrace $, and the first equation in (*) is satisfied in the sense of distributions.

In addition we discuss the possible application of our results for mathematical modelling of some phenomena in cell biology.

Abstract in PDF.