Past PDE CDT Lunchtime Seminar

9 March 2017
12:00
Abstract

Consider a family of uniformly bounded $W^{2,p}$ isometric immersions of an $n$-dimensional (semi-) Riemannian manifold into (resp., semi-) Euclidean spaces. Are the weak limits still isometric immersions?

We answer the question in the affirmative for $p>n$ in the Riemannian case, by exploiting the div-curl structure of the Gauss-Codazzi-Ricci equations, which describe the curvature flatness of the isometric immersions. Along the way a generalised div-curl lemma in Banach spaces is established. Moreover, the endpoint case $p=n=2$ is settled. 

In the semi-Riemannian case we reduce the problem to the weak continuity of H. Cartan's structural equations in $W^{1,p}_{\rm loc}$, which is proved by a generalised compensated compactness theorem relating the weak continuity of quadratic forms to the principal symbols of differential constraints. Again for $p>n$ we obtain the weak rigidity. The case of degenerate hypersurfaces are also discussed, as well as connections to PDEs in fluid dynamics.

  • PDE CDT Lunchtime Seminar
2 March 2017
12:00
Moritz Kassmann
Abstract

We report on recent developments in the study of nonlocal operators. The central object of the talk are quadratic forms similar to those that define Sobolev spaces of fractional order. These objects are naturally linked to Markov processes via the theory of Dirichlet forms. We provide regularity results for solutions to corresponding integrodifferential equations. Our emphasis is on forms with singularand anisotropic measures. Some of the objects under consideration are related to the Boltzmann equation, which leads to an interesting question of comparability of quadrativ forms. The talk is based on recent results joint with B. Dyda and with K.-U. Bux and T. Schulze.

  • PDE CDT Lunchtime Seminar
23 February 2017
12:00
Peter Bella
Abstract
Wrinkling of thin elastic sheets can be viewed as a way how to avoid compressive stresses. While the question of where the wrinkles appear is well-understood, understanding properties of wrinkling is not trivial. Considering a variational viewpoint, the problem amounts to minimization of an elastic energy, which can be viewed as a non-convex membrane energy singularly perturbed by a higher-order bending term. To understand the global minimizer (ground state), the first step is to identify its energy, in particular its dependence on the small physical parameter (thickness). I will discuss several problems where the optimal energy scaling law was identified.
 
  • PDE CDT Lunchtime Seminar
16 February 2017
12:00
Sarah Penington
Abstract


The non-local Fisher KPP equation is used to model non-local interaction and competition in a population. I will discuss recent work on solutions of this equation with a compactly supported initial condition, which strengthens results on the spreading speed obtained by Hamel and Ryzhik in 2013. The proofs are probabilistic, using a Feynman-Kac formula and some ideas from Bramson's 1983 work on the (local) Fisher KPP equation.

  • PDE CDT Lunchtime Seminar
9 February 2017
12:00
Abstract
Of concern is the regularity of solutions to the classical water wave problem for two-dimensional Euler flows with vorticity. It is shown that the profile together with all streamlines beneath a periodic water wave travelling over a flat bed are real-analytic curves, provided that the vorticity function is merely integrable and that there are no stagnation points in the flow. It is furthermore exposed that the analyticity of streamlines can be used to characterise intrinsically symmetric water waves. 
  • PDE CDT Lunchtime Seminar
2 February 2017
12:00
Abstract

Systems that have more than one conserved quantity (i.e. energy plus momentum, density etc.), can exhibit quite interesting temperature profiles in non-equilibrium stationary states. I will present some numerical experiment and mathematical result. I will also expose some other connected problems, always concerning thermal boundary conditions in hydrodynamic limits.
 

  • PDE CDT Lunchtime Seminar
26 January 2017
12:00
Jan Burczak
Abstract

Patlak-Keller-Segel equations 
\[
\begin{aligned}
u_t - L u &= - \mathop{\text{div}\,} (u \nabla v) \\
v_t - \Delta v &= u,
\end{aligned}
\]
where L is a dissipative operator, stem from mathematical chemistry and mathematical biology.
Their variants describe, among others, behaviour of chemotactic populations, including feeding strategies of zooplankton or of certain insects. Analytically, Patlak-Keller-Segel equations reveal quite rich dynamics and a delicate global smoothness vs. blowup dichotomy. 
We will discuss smoothness/blowup results for popular variants of the equations, focusing on the critical cases, where dissipative and aggregative forces seem to be in a balance. A part of this talk is based on joint results with Rafael Granero-Belinchon (Lyon).

  • PDE CDT Lunchtime Seminar
7 December 2016
16:00
Abstract

We prove higher differentiability of bounded local minimizers to some degenerate functionals satisfying anisotropic growth conditions. In the two-dimensional case we also study the Lipschitz regularity of such minimizers without any limitation on the exponents of anisotropy.

  • PDE CDT Lunchtime Seminar
24 November 2016
12:00
Sebastian Schwarzacher
Abstract
I will present a new result which was established in collaboration with M. Bulıcek and J. Burczak. We established an existence, uniqueness and optimal regularity results for very weak solutions to certain incompressible non-Newtonian fluids. We introduce structural assumptions of Uhlenbeck type on the stress tensor. These as-sumptions are sufficient and to some extend also necessary to built a unified theory. Our approach leads qualitatively to the same so called Lp-theory as the one that is available for the linear Stokes equation.
  • PDE CDT Lunchtime Seminar

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