Past Seminars › Seminar Series

Partial Differential Equations Seminar (past)

Mon, 13/05
17:00 		Gustav Holzegel (Imperial College London) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
The Wave Equation on Asymptotically Anti de Sitter Black Hole Spacetimes
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, 06/05
17:00 		Eduard Feireisl (institute of mathematics of the Academy of sciences of the Czech Republic) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Multiple scales in the dynamics of compressible fluids
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.
Fri, 03/05
17:00 		Mikhail Korobkov (Sobolev Institute of Mathematics, Novosibirsk) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
The Morse-Sard Theorem for $ W^{n,1} $ Sobolev functions on $ \mathbb R^n $ and applications in fluid mechanics
The talk is based on the joint papers [Bourgain J., Korobkov M.V. and Kristensen~J.: Journal fur die reine und angewandte Mathematik (Crelles Journal). DOI: 10.1515/crelle-2013-0002]  and  [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.
Mon, 29/04
15:00 		Willi Jaeger (Heidelberg University) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
INTERACTIONS OF THE FLUID AND SOLID PHASES IN COMPLEX MEDIA - COUPLING REACTIVE FLOWS, TRANSPORT AND MECHANICS
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.
Mon, 22/04
17:00 		Rolando Magnanini (Università degli Studi di Firenze) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Time-invariant surfaces in evolution equations
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, 25/02
17:00 		Lars Andersson (Max Planck Institute for Gravitational Physics) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Self-gravitating elastic bodies
Self-gravitating elastic bodies provide models for extended objects in general relativity. I will discuss constructions of static and rotating self-gravitating bodies, as well as recent results on the initial value problem for self-gravitating elastic bodies.
Mon, 18/02
17:00 		Raphaël Danchin (Université Paris Est) Partial Differential Equations Seminar Add to calendar Gibson Grd floor SR
A Lagrangian approach for nonhomogeneous incompressible fluids
In this talk we focus on the incompressible Navier–Stokes equations with variable density. The aim is to prove existence and uniqueness results in the case of a discontinuous initial density (typically we are interested in discontinuity along an interface). In the first part of the talk, by making use of Fourier analysis techniques, we establish the existence of global-in-time unique solutions in a critical functional framework, under some smallness condition over the initial data, In the second part, we use another approach to avoid the smallness condition over the nonhomogeneity : as a matter of fact, one may consider any density bounded and bounded away from zero and still get a unique solution. The velocity is required to have subcritical regularity, though. In all the talk, the Lagrangian formulation for describing the flow plays a key role in the analysis.
Mon, 11/02
17:00 		Fabricio Macià Lang (Universidad Politécnica de Madrid) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Defect measures and Schrödinger flows
Defect measures have successfully been used, in a variety of contexts, as a tool to quantify the lack of compactness of bounded sequences of square-integrable functions due to concentration and oscillation effects. In this talk we shall present some results on the structure of the set of possible defect measures arising from sequences of solutions to the linear Schrödinger equation on a compact manifold. This is motivated by questions related to understanding the effect of geometry on dynamical aspects of the Schrödinger flow, such as dispersive effects and unique continuation. It turns out that the answer to these questions depends strongly on global properties of the geodesic flow on the manifold under consideration: this will be illustrated by discussing with a certain detail the examples of the the sphere and the (flat) torus.
Mon, 04/02
17:00 		S. V. Kislyakov (V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Differential expressions with mixed homogeneity and spaces of smooth functions they generate
Let $ {T_1,...,T_l} $ be a collection of differential operators with constant coefficients on the torus $ \mathbb{T}^n $. Consider the Banach space $ X $ of functions $ f $ on the torus for which all functions $ T_j f $, $ j=1,...,l $, are continuous. The embeddability of $ X $ into some space $ C(K) $ as a complemented subspace will be discussed. The main result is as follows. Fix some pattern of mixed homogeneity and extract the senior homogeneous parts (relative to the pattern chosen) $ {\tau_1,...,\tau_l} $ from the initial operators $ {T_1,...,T_l} $. If there are two nonproportional operators among the $ \tau_j $ (for at least one homogeneity pattern), then $ X $ is not isomorphic to a complemented subspace of $ C(K) $ for any compact space $ K $. The main ingredient of the proof is a new Sobolev-type embedding theorem. It generalises the classical embedding of $ {\stackrel{\circ}{W}}_1^1(\mathbb{R}^2) $ to $ L^2(\mathbb{R}^2) $. The difference is that now the integrability condition is imposed on certain linear combinations of derivatives of different order of several functions rather than on the first order derivatives of one function. This is a joint work with D. Maksimov and D. Stolyarov.
Mon, 28/01
17:00 		John M. Ball (Oxford) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Hadamard's compatibility condition for microstructures

The talk will discuss generalizations of the classical Hadamard jump  condition to general locally Lipschitz maps, and applications to
polycrystals. This is joint work with Carsten Carstensen.
Mon, 21/01
17:00 		Qian Wang (OxPDE, University of Oxford) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Rough Solutions of Einstein Vacuum equations in CMCSH gauge
I will report my work on rough solutions to Cauchy problem for the Einstein vacuum equations in CMC spacial harmonic gauge, in which we obtain the local well-posedness result in $ H^s $, $ s $>$ 2 $. The novelty of this approach lies in that, without resorting to the standard paradifferential regularization over the rough, Einstein metric $ \bf{g} $, we manage to implement the commuting vector field approach to prove Strichartz estimate for geometric wave equation $ \Box_{\bf{g} } \phi=0 $ directly. If time allows, I will talk about my work in progress on the sharp results for the more general quasilinear wave equations by vector fields approach.
Mon, 14/01
17:00 		Jonathan Bevan (University of Surrey) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
N-covering stationary points and constrained variational problems
In this talk we show how degree N maps of the form $ u_{N}(z) = \frac{z^{N}}{|z|^{N-1}} $ arise naturally as stationary points of functionals like the Dirichlet energy. We go on to show that the $ u_{N} $ are minimizers of related variational problems, including one whose associated Euler-Lagrange equation bears a striking resemblance to a system studied by N. Meyers in the 60s, and another where the constraint $ \det \nabla u = 1 $ a.e. plays a prominent role.
Mon, 19/11/2012
15:45 		Michael Westdickenberg (University of Aachen) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
A variational time discretization for the compressible Euler equations
Mon, 12/11/2012
17:00 		Ivo Kaelin (with D. Christodoulou) (ETH Zurich) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Crystalline solids with a uniform distribution of dislocations
Crystalline solids are descibed by a material manifold endowed with a certain structure which we call crystalline. This is characterized by a canonical 1-form, the integral of which on a closed curve in the material manifold represents, in the continuum limit, the sum of the Burgers vectors of all the dislocation lines enclosed by the curve. In the case that the dislocation distribution is uniform, the material manifold becomes a Lie group upon the choice of an identity element. In this talk crystalline solids with uniform distributions of the two elementary kinds of dislocations, edge and screw dislocations, shall be considered. These correspond to the two simplest non-Abelian Lie groups, the affine group and the Heisenberg group respectively. The statics of a crystalline solid are described in terms of a mapping from the material domain into Euclidean space. The equilibrium configurations correspond to mappings which minimize a certain energy integral. The static problem is solved in the case of a small density of dislocations.
Mon, 29/10/2012
17:00 		Carsten Carstensen (Humboldt Universität zu Berlin) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Five Trends in the Mathematical Foundation of Computational PDEs
This presentation concerns five topics in computational partial differential equations with the overall goals of reliable error control and efficient simulation. The presentation is also an advertisement for nonstandard discretisations in linear and nonlinear Computational PDEs with surprising advantages over conforming finite element schemes and the combination of the two. The equivalence of various first-order methods is explained for the linear Poisson model problem with conforming (CFEM), nonconforming (NC-FEM), and mixed finite element methods (MFEM) and others discontinuous Galerkin finite element (dGFEM). The Stokes equations illustrate the NCFEM and the pseudo-stress MFEM and optimal convergence of adaptive mesh-refining as well as for guaranteed error bounds. An optimal adaptive CFEM computation of elliptic eigenvalue problems and the computation of guaranteed upper and lower eigenvalue bounds based on NCFEM. The obstacle problem and its guaranteed error control follows another look due to D. Braess with guaranteed error bounds and their effectivity indices between 1 and 3. Some remarks on computational microstructures with degenerate convex minimisation problems conclude the presentation.
Mon, 22/10/2012
17:00 		Juha Kinnunen (Aalto University) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
On the definition and properties of superparabolic functions
We review potential theoretic aspects of degenerate parabolic PDEs of p-Laplacian type. Solutions form a similar basis for a nonlinear parabolic potential theory as the solutions of the heat equation do in the classical theory. In the parabolic potential theory, the so-called superparabolic functions are essential. For the ordinary heat equation we have supercaloric functions. They are defined as lower semicontinuous functions obeying the comparison principle. The superparabolic functions are of actual interest also because they are viscosity supersolutions of the equation. We discuss their existence, structural, convergence and Sobolev space properties. We also consider the definition and properties of the nonlinear parabolic capacity and show that the infinity set of a superparabolic function is of zero capacity.
Mon, 15/10/2012
17:00 		Karine Beauchard (CNRS and Ecole Polytechnique) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Control of bilinear systems
Mon, 08/10/2012
17:00 		José Antonio Carrillo de la Plata ((Imperial College) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Blow-up & Stationary States
We will discuss how optimal transport tools can be used to analyse the qualitative behavior of continuum systems of interacting particles by fully attractive or short-range repulsive long-range attractive potentials.

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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