Fri, 03 May 2013

17:00 - 18:00
Gibson 1st Floor SR

### The Morse-Sard Theorem for $W^{n,1}$ Sobolev functions on $\mathbb R^n$ and applications in fluid mechanics

Mikhail Korobkov
(Sobolev Institute of Mathematics, Novosibirsk)
Abstract
The talk is based on the joint papers [{\it Bourgain J., Korobkov

M.V. and Kristensen~J.}: Journal fur die reine und angewandte Mathematik

(Crelles

Journal).

DOI: 10.1515/crelle-2013-0002] \ and \

[{\it Korobkov~M.V., Pileckas~K. and Russo~R.}:

arXiv:1302.0731, 4 Feb 2013]

We establish Luzin $N$ and Morse--Sard

properties for functions from the Sobolev space

$W^{n,1}(\mathbb R^n)$. Using these results we prove

that almost all level sets are finite disjoint unions of

$C^1$-smooth compact manifolds of dimension

$n-1$. These results remain valid also within

the larger space of functions of bounded variation

$BV_n(\mathbb R^n)$.

As an application, we study the nonhomogeneous boundary value problem

for the Navier--Stokes equations of steady motion of a viscous

incompressible fluid in arbitrary bounded multiply connected

plane or axially-symmetric spatial domains. We prove that this

problem has a solution under the sole necessary condition of zero total

flux through the boundary.

The problem was formulated by Jean Leray 80 years ago.

The proof of the main result uses Bernoulli's law

for a weak solution to the Euler equations based on the above-mentioned

Morse-Sard property for Sobolev functions.

Thu, 02 May 2013
12:00
Gibson 1st Floor SR

### Partial Regularity for constrained minimisers of quasi convex functionals with $p$-growth

Christopher Hopper
(OxPDE, University of Oxford)
Abstract

We consider minimisers of integral functionals $F$ over a ‘constrained’ class of $W^{1,p}$-mappings from a bounded domain into a compact Riemannian manifold $M$, i.e. minimisers of $F$ subject to holonomic constraints. Integrands of the form $f(Du)$ and the general $f(x,u,Du)$ are considered under natural strict $p$-quasiconvexity and $p$-growth assumptions for any exponent $1 < p <+\infty$. Unlike the harmonic and $p$-harmonic map case, the quasiconvexity condition requires one to linearise the map at the level of the gradient. In a bid to give a direct proof of partial $C^{1,α}-regularity for such minimisers, we developing an appropriate notion of a tangential harmonic approximation together with a discussion on the difficulties in establishing Caccioppoli-type inequalities. The need in the latter problem to construct suitable competitors to the minimiser via the so-called Luckhaus Lemma presents difficulties quite separate to that of the unconstrained case. We will give a proof of this lemma together with a discussion on the implications for higher integrability. Thu, 06 Jun 2013 12:00 - 13:00 Gibson 1st Floor SR ### Numerical approximations for a nonloncal model for sandpiles Mayte Pérez-Llanos (Universidad Autonoma de Madrid) Abstract In this talk we study numerical approximations of continuous solutions to a nonlocal$p$-Laplacian type diffusion equation, $u_t (t, x) = \int_\Omega J(x − y)|u(t, y) − u(t, x)|^{p-2} (u(t, y) − u(t, x)) dy.$ First, we find that a semidiscretization in space of this problem gives rise to an ODE system whose solutions converge uniformly to the continuous one, as the mesh size goes to zero. Moreover, the semidiscrete approximation shares some properties with the continuous problem: it preserves the total mass and the solution converges to the mean value of the initial condition, as$t$goes to infinity. Next, we discretize also the time variable and present a totally discrete method which also enjoys the above mentioned properties. In addition, we investigate the limit as$p\$ goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile.
Finally, we present some numerical experiments that illustrate our results.
This is a joint work with J. D. Rossi.
Mon, 20 May 2013

17:00 - 18:00
Gibson 1st Floor SR

### Analysis of some nonlinear PDEs from multi-scale geophysical applications

Bin Cheng
(University of Surrey)
Abstract
This talk is regarding PDE systems from geophysical applications with multiple time scales, in which linear skew-self-adjoint operators of size 1/epsilon gives rise to highly oscillatory solutions. Analysis is performed in justifying the limiting dynamics as epsilon goes to zero; furthermore, the analysis yields estimates on the difference between the multiscale solution and the limiting solution. We will introduce a simple yet effective time-averaging technique which is especially useful in general domains where Fourier analysis is not applicable.
Mon, 13 May 2013

17:00 - 18:00
Gibson 1st Floor SR

### The Wave Equation on Asymptotically Anti de Sitter Black Hole Spacetimes

Gustav Holzegel
(Imperial College London)
Abstract
The study of wave equations on black hole backgrounds provides important insights for the non-linear stability problem for black holes. I will illustrate this in the context of asymptotically anti de Sitter black holes and present both stability and instability results. In particular, I will outline the main ideas of recent work with J. Smulevici (Paris) establishing a logarithmic decay in time for solutions of the massive wave equation on Kerr-AdS black holes and proving that this slow decay rate is in fact sharp.
Mon, 06 May 2013

17:00 - 18:00
Gibson 1st Floor SR

### Multiple scales in the dynamics of compressible fluids

Eduard Feireisl
(institute of mathematics of the Academy of sciences of the Czech Republic)
Abstract
We discuss several singular limits for a scaled system of equations

(barotropic Navier-Stokes system), where the characteristic numbers become

small or infinite''. In particular, we focus on the situations relevant

in certain geophysical models with low Mach, large Rossby and large

Reynolds numbers. The limit system is rigorously identified in the

framework of weak solutions. The relative entropy inequality and careful

analysis of certain oscillatory integrals play crucial role.

Mon, 22 Apr 2013

17:00 - 18:00
Gibson 1st Floor SR

### Time-invariant surfaces in evolution equations

Rolando Magnanini
(Università degli Studi di Firenze)
Abstract
A time-invariant level surface is a (codimension one)

spatial surface on which, for every fixed time, the solution of an

evolution equation equals a constant (depending on the time). A

relevant and motivating case is that of the heat equation. The

occurrence of one or more time-invariant surfaces forces the solution

to have a certain degree of symmetry. In my talk, I shall present a

set of results on this theme and sketch the main ideas involved, that

intertwine a wide variety of old and new analytical and geometrical

techniques.

Mon, 29 Apr 2013

15:00 - 16:00
Gibson 1st Floor SR

### INTERACTIONS OF THE FLUID AND SOLID PHASES IN COMPLEX MEDIA - COUPLING REACTIVE FLOWS, TRANSPORT AND MECHANICS

Willi Jaeger
(Heidelberg University)
Abstract
Modelling reactive flows, diffusion, transport and mechanical interactions in media consisting of multiple phases, e.g. of a fluid and a solid phase in a porous medium, is giving rise to many open problems for multi-scale analysis and simulation. In this lecture, the following processes are studied:

diffusion, transport, and reaction of substances in the fluid and the solid phase,

mechanical interactions of the fluid and solid phase,

change of the mechanical properties of the solid phase by chemical reactions,

volume changes (“growth”) of the solid phase.

These processes occur for instance in soil and in porous materials, but also in biological membranes, tissues and in bones. The model equations consist of systems of nonlinear partial differential equations, with initial-boundary conditions and transmission conditions on fixed or free boundaries, mainly in complex domains. The coupling of processes on different scales is posing challenges to the mathematical analysis as well as to computing. In order to reduce the complexity, effective macroscopic equations have to be derived, including the relevant information from the micro scale.

In case of processes in tissues, a homogenization limit leads to an effective, mechanical system, containing a pressure gradient, which satisfies a generalized, time-dependent Darcy law, a Biot-law, where the chemical substances satisfy diffusion-transport-reaction equations and are influencing the mechanical parameters.

The interaction of the fluid and the material transported in a vessel with its flexible wall, incorporating material and changing its structure and mechanical behavior, is a process important e.g. in the vascular system (plague-formation) or in porous media.

The lecture is based on recent results obtained in cooperation with A. Mikelic, M. Neuss-Radu, F. Weller and Y. Yang.

Thu, 09 May 2013
12:01
Gibson 1st Floor SR

### Weak solutions to the barotropic Navier-Stokes system with slip boundary conditions in time dependent domains and incompressible limits

Šárka Nečasová
(Academy of Sciences of the Czech Republic)
Abstract
We consider the compressible (barotropic) Navier-Stokes system on time-dependent domains, supplemented with slip boundary conditions. Our approach is based on penalization of the boundary behaviour, viscosity, and the pressure in the weak formulation. Global-in-time weak solutions are obtained. Secondly, we suppose that the characteristic speed of the ﬂuid is domi- nated by the speed of sound and perform the low Mach number limit in the framework of weak solutions. The standard incompressible Navier-Stokes system is identiﬁed as the target problem. References:
[1] E. Feireisl, O. Kreml, S. Nečasová, J. Neustupa, and J. Stebel. Weak solutions to the barotropic NavierStokes system with slip boundary conditions in time dependent domains. J. Diﬀerential Equations, 254:125–140, 2013.
[2] E. Feireisl, O. Kreml, S. Nečasová, J. Neustupa, and J. Stebel. Incompressible limits of ﬂuids excited by moving boundaries. Submitted
Tue, 12 Mar 2013
14:30
Gibson 1st Floor SR

### Twistor Diagrams

Andrew Hodges
(Oxford)
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