Forthcoming Seminars › Seminar Series

Partial Differential Equations Seminar

Mon, 11/10/2010
17:00 		Georg Dolzmann (Universitaet Regensburg) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Modeling and simulation of vectorfields on membranes
The fundamental models for lipid bilayers are curvature based and neglect the internal structure of the lipid layers. In this talk, we explore models with an additional order parameter which describes the orientation of the lipid molecules in the membrane and compare their predictions based on numerical simulations. This is joint work with Soeren Bartels (Bonn) and Ricardo Nochetto (College Park).
Mon, 18/10/2010
17:00 		Alexis Vasseur (University of Oxford) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Relative entropy method applied to the stability of shocks for systems of conservation laws

We develop a theory based on relative entropy to show stabilityand uniqueness of extremal entropic Rankine-Hugoniot discontinuities forsystems of conservation laws (typically 1-shocks, n-shocks, 1-contactdiscontinuities and n-contact discontinuities of big amplitude), amongbounded entropic weak solutions having an additional strong traceproperty. The existence of a convex entropy is needed. No BV estimateis needed on the weak solutions considered. The theory holds withoutsmallness condition. The assumptions are quite general. For instance, thestrict hyperbolicity is not needed globally. For fluid mechanics, thetheory handles solutions with vacuum.
Mon, 25/10/2010
17:00 		Luigi Berselli (Universita di Pisa) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
On averaged equations for turbulent flows
I will make a short review of some continous approximations to the Navier-Stokes equations, especially with the aim of introducing alpha models for the Large Eddy Simulation of turbulent flows. Next, I will present some recent results about approximate deconvolution models, derived with ideas similar to image processing. Finally, I will show the rigorous convergence of solutions towards those of the averaged fluid equations.
Mon, 01/11/2010
17:00 		Petru Mironescu (Universite Lyon 1) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
What is a circle-valued map made of?
The maps $ u $ which are continuous in $ {\mathbb R}^n $ and circle-valued are precisely the maps of the form $ u=\exp (i\varphi) $, where the phase $ \varphi $ is continuous and real-valued. In the context of Sobolev spaces, this is not true anymore: a map $ u $ in some Sobolev space $ W^{s,p} $ need not have a phase in the same space. However, it is still possible to describe all the circle-valued Sobolev maps. The characterization relies on a factorization formula for Sobolev maps, involving three objects: good phases, bad phases, and topological singularities. This formula is the analog, in the circle-valued context, of Weierstrass' factorization theorem for holomorphic maps. The purpose of the talk is to describe the factorization and to present a puzzling byproduct concerning sums of Dirac masses.
Mon, 08/11/2010
17:00 		Konstantin Pileckas (Vilnius University) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
On the stationary Navier-Stokes system with nonhomogeneous boundary data
We study the nonhomogeneous boundary value problem for the Navier–Stokes equations
\[ \left\{ \begin{array}{rcl} -\nu \Delta{\bf u}+\big({\bf u}\cdot \nabla\big){\bf u} +\nabla p&=\qquad \hbox{\rm in }\;\;\Omega,\\[4pt] {\rm div}\,{\bf u}&=&0 \qquad \hbox{\rm in }\;\;\Omega,\\[4pt] {\bf u}&=\qquad \hbox{\rm on }\;\;\partial\Omega \end{array}\right \eqno(1) \]
in a bounded multiply connected domain $ \Omega\subset\mathbb{R}^n $ with the boundary $ \partial\Omega $, consisting of $ N $ disjoint components $ \Gamma_j $. Starting from the famous J. Leray's paper published in 1933, problem (1) was a subject of investigation in many papers. The continuity equation in (1) implies the necessary solvability condition
$$ \int\limits_{\partial\Omega}{\bf a}\cdot{\bf n}\,dS=\sum\limits_{j=1}^N\int\limits_{\Gamma_j}{\bf a}\cdot{\bf n}\,dS=0,\eqno(2) $$
where $ {\bf n} $ is a unit vector of the outward (with respect to $ \Omega $) normal to $ \partial\Omega $. However, for a long time the existence of a weak solution $ {\bf u}\in W^{1,2}(\Omega) $ to problem (1) was proved only under the stronger condition
$$ {\cal F}_j=\int\limits_{\Gamma_j}{\bf a}\cdot{\bf n}\,dS=0,\qquad j=1,2,\ldots,N. \eqno(3) $$
During the last 30 years many partial results concerning the solvability of problem (1) under condition (2) were obtained. A short overview of these results and the detailed study of problem (1) in a two–dimensional bounded multiply connected domain $ \Omega=\Omega_1\setminus\Omega_2, \;\overline\Omega_2\subset \Omega_1 $ will be presented in the talk. It will be proved that this problem has a solution, if the flux $ {\cal F}=\int\limits_{\partial\Omega_2}{\bf a}\cdot{\bf n}\,dS $ of the boundary datum through $ \partial\Omega_2 $ is nonnegative (outflow condition).
Mon, 15/11/2010
17:00 		Lisa Beck (Scuola Normale Superiore di Pisa) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
The role of small space dimensions in the regularity theory of elliptic problems
Let $ u \in W^{1,p}(\Omega,\R^N) $, $ \Omega $ a bounded domain in $ \R^n $, be a minimizer of a convex variational integral or a weak solution to an elliptic system in divergence form. In the vectorial case, various counterexamples to full regularity have been constructed in dimensions $ n \geq 3 $, and it is well known that only a partial regularity result can be expected, in the sense that the solution (or its gradient) is locally continuous outside of a negligible set. In this talk, we shall investigate the role of the space dimension $ n $ on regularity: In arbitrary dimensions, the best known result is partial regularity of the gradient $ Du $ (and hence for $ u $) outside of a set of Lebesgue measure zero. Restricting ourselves to the partial regularity of $ u $ and to dimensions $ n \leq p+2 $, we explain why the Hausdorff dimension of the singular set cannot exceed $ n-p $. Finally, we address the possible existence of singularities in two dimensions.
Mon, 22/11/2010
17:00 		Jose Carillo de la Plata (Universitat Autònoma de Barcelona) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Keller-Segel, Fast-Diffusion and Functional Inequalities
It will be shown how the critical mass classical Keller-Segel system and the critical displacement convex fast-diffusion equation in two dimensions are related. On one hand, the critical fast diffusion entropy functional helps to show global existence around equilibrium states of the critical mass Keller-Segel system. On the other hand, the critical fast diffusion flow allows to show functional inequalities such as the Logarithmic HLS inequality in simple terms who is essential in the behavior of the subcritical mass Keller-Segel system. HLS inequalities can also be recovered in several dimensions using this procedure. It is crucial the relation to the GNS inequalities obtained by DelPino and Dolbeault. This talk corresponds to two works in preparation together with E. Carlen and A. Blanchet, and with E. Carlen and M. Loss.
Mon, 29/11/2010
17:00 		Endre Suli (University of Oxford) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Navier-Stokes-Fokker-Planck systems in kinetic models of dilute polymers: existence and equilibration of global weak solutions
We show the existence of global-in-time weak solutions to a general class of bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three space dimensions for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum for the Navier-Stokes equation and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian of the model, we prove the existence of a global-in-time weak solution to the coupled Navier-Stokes-Fokker-Planck system. It is also shown that in the absence of a body force, the weak solution decays exponentially in time to the equilibrium solution, at a rate that is independent of the choice of the initial datum and of the centre-of-mass diffusion coefficient. The talk is based on joint work with John W. Barrett [Imperial College London].

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