Past PDE CDT Lunchtime Seminar

18 May 2017
12:00
Abstract

We determine the hydrodynamic limit of some kinetic equations with either stochastic Vlasov force term or stochastic collision kernel. We obtain stochastic second-order parabolic equations at the limit. In the regime we consider, we also observe (or do not observe) some phenomena of enhanced diffusion. Joint works with Nils Caillerie, Arnaud Debussche, Martina Hofmanová.
 

  • PDE CDT Lunchtime Seminar
27 April 2017
12:00
Abstract

We consider the Euler–Voigt equations in a bounded domain as an approximation for the 3D Euler equations. We adopt suitable physical conditions and show that the solutions of the Voigt equations are global, do not smooth out the solutions and converge to the solutions of the Euler equations, hence they represent a good model.

  • PDE CDT Lunchtime Seminar
7 April 2017
12:00
Abstract

We study vortex sheets for the relativistic Euler equations in three-dimensional Minkowski spacetime. The problem is a nonlinear hyperbolic problem with a characteristic free boundary. The so-called Lopatinskii condition holds only in a weak sense, which yields losses of derivatives. A necessary condition for the weak stability is obtained by analyzing roots of the Lopatinskii determinant associated to the linearized problem. Under such stability condition,  we prove short-time existence and nonlinear stability of relativistic vortex sheets by the Nash-Moser iterative scheme.

  • PDE CDT Lunchtime Seminar
7 April 2017
11:00
Paolo Secchi
Abstract

We consider the free boundary problem for 2D current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. 
In this talk we present our results about the well-posedness of the problem in the sense of Hadamard, under a suitable stability condition, that is the 
local-in-time existence in Sobolev spaces and uniqueness of smooth solutions to the Cauchy problem, and the strong continuous dependence on the data in the same topology.
Joint works with: Alessandro Morando and Paola Trebeschi.
 

  • 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

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