Forthcoming Seminars › Seminar Series

Partial Differential Equations Seminar

Mon, 13/10/2008
17:00 		Gregory Seregin (Oxford) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Liouville type theorems for Navier-Stokes equations
In the lecture, I am going to explain a connection between local regularity theory for the Navier-Stokes equations and Liouville type theorems for bounded ancient solutions to these equations.
Mon, 20/10/2008
17:00 		Christopher Jones (University of North Carolina & Warwick) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
"Oscillations" and Stability in Multi-Dimensions
Mon, 27/10/2008
17:00 		Peter Constantin (Chicago) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
On the zero temperature limit of interacting corpora
I will talk about recent work concerning the Onsager equation on metric spaces. I will describe a framework for the study of equilibria of melts of corpora – bodies with finitely many degrees of freedom, such as stick-and-ball models of molecules.
Mon, 03/11/2008
17:00 		Philippe Laurençot (Toulouse) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Critical mass in generalized Smoluchowski-Poisson equations
In space dimension 2, it is well-known that the Smoluchowski-Poisson system (also called the simplified or parabolic-elliptic Keller-Segel chemotaxis model) exhibits the following phenomenon: there is a critical mass above which all solutions blow up in finite time while all solutions are global below that critical mass. We will investigate the case of the critical mass along with the stability of self-similar solutions with lower masses. We next consider a generalization to several space dimensions which involves a nonlinear diffusion and show that a similar phenomenon takes place but with some different features.
Mon, 10/11/2008
17:00 		Demetrios Christodoulou (ETH Zurich) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
The formation of shocks in 3-dimensional fluids
Mon, 17/11/2008
17:00 		Geneviève Raugel (Université Paris Sud) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
A hyperbolic pertubation of the Navier-Stokes equations
Y. Brenier, R. Natalini and M. Puel have considered a “relaxation" of the Euler equations in R2. After an approriate scaling, they have obtained the following hyperbolic version of the Navier-Stokes equations, which is similar to the hyperbolic version of the heat equation introduced by Cattaneo,
$$\varepsilon u_{tt}^\varepsilon + u_t^\varepsilon -\Delta u^\varepsilon +P (u^\varepsilon \nabla u^\varepsilon) \, = \, Pf~, \quad (u^\varepsilon(.,0), u_t^\varepsilon(.,0)) \, = \, (u_0(.),u_1(.))~, \quad (1) $$
where $ P $ is the classical Leray projector and $ \varepsilon $ is a small, positive number. Under adequate hypotheses on the forcing term $ f $, we prove global existence and uniqueness of a mild solution $ (u^\varepsilon,u_t^\varepsilon) \in C^0([0, +\infty), H^{1}({\bf R}^2) \times L^2({\bf R}^2)) $ of (1), for large initial data $ (u_0,u_1) $ in $ H^{1}({\bf R}2) \times L^2({\bf R}2) $, provided that $ \varepsilon>0 $ is small enough, thus improving the global existence results of Brenier, Natalini and Puel (actually, we can work in less regular Hilbert spaces). The proof uses appropriate Strichartz estimates, combined with energy estimates. We also show that $ (u^\varepsilon,u_t^\varepsilon) $ converges to $ (v,v_t) $ on finite intervals of time $ [t_0,t_1] $, $ 0 <+ \infty $, when $ \varepsilon $ goes to $ 0 $, where $ v $ is the solution of the corresponding Navier-Stokes equations
$$ v_t -\Delta v +P (v\nabla v) \, = \, Pf~, \quad v(.,0) \, = \, u_0~. \quad (2) $$
We also consider Equation (1) in the three-dimensional case. Here we expect global existence results for small data. Under appropriate assumptions on the forcing term, we prove global existence and uniqueness of a mild solution $ (u^\varepsilon,u_t^\varepsilon) \in C^0([0, +\infty), H^{1+\delta}({\bf R}^3) \times H^{\delta}({\bf R}^3)) $ of (1), for initial data $ (u_0,u_1) $ in $ H^{1 +\delta}({\bf R}^3) \times H^{\delta}({\bf R}^3) $ (where $ \delta >0 $ is a small positive number), provided that $ \varepsilon > 0 $ is small enough and that $ u_0 $ and $ f $ satisfy a smallness condition. (Joint work with Marius Paicu)
Tue, 25/11/2008
15:00 		Georg Dolzmann (University of Regensburg) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Relaxation and Gamma convergence results in models for crystal plasticity
Mon, 01/12/2008
13:00 		Isaac Vikram Chenchiah (University of Bristol) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Strain and stress fields in shape-memory and rigid-perfectly plastic polycrystals
he study of polycrystals of shape-memory alloys and rigid-perfectly plastic materials gives rise to problems of nonlinear homogenization involving degenerate energies. We present a characterisation of the strain and stress fields for some classes of problems in plane strain and also for some three-dimensional situations. Consequences for shape-memory alloys and rigid-perfectly plastic materials are discussed through model problems. In particular we explore connections to previous conjectures characterizing those shape-memory polycrystals with non-trivial recoverable strain.

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