Forthcoming Seminars › Seminar Series

Partial Differential Equations Seminar

Mon, 17/01/2011
17:00 		Jonathan Ben-Artzi (Brown University) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Linear instability of the Relativistic Vlasov-Maxwell system
We consider the Relativistic Vlasov-Maxwell system of equations which describes the evolution of a collisionless plasma. We show that under rather general conditions, one can test for linear instability by checking the spectral properties of Schrodinger-type operators that act only on the spatial variable, not the full phase space. This extends previous results that show linear and nonlinear stability and instability in more restrictive settings.
Mon, 24/01/2011
17:00 		Jean-Yves Chemin (Universite Pierre et Marie Curie) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Slowly varying in one direction global solution of the incompressible Navier-Stokes system
The purpose of this talk is to provide a large class of examples of large initial data which gives rise to a global smooth solution. We shall explain what we mean by large initial data. Then we shall explain the concept of slowly varying function and give some flavor of the proofs of global existence.
Mon, 31/01/2011
17:00 		Edriss Titi (University of California) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
On the Loss of Regularity for the Three-Dimensional Euler Equations

A basic example of  shear flow wasintroduced  by DiPerna and Majda to study the weaklimit of oscillatory solutions of the Eulerequations of incompressible ideal fluids. Inparticular, they proved by means of this examplethat weak limit of solutions of Euler equationsmay, in some cases, fail to be a solution of Eulerequations. We use this shear flow example toprovide non-generic, yet nontrivial, examplesconcerning the immediate loss of smoothness andill-posedness of solutions of the three-dimensionalEuler equations, for initial data that do notbelong to $C^{1,\alpha}$. Moreover, we show bymeans of this shear flow example the existence ofweak solutions for the three-dimensional Eulerequations with vorticity that is  having anontrivial density concentrated on non-smoothsurface. This is very different from what has beenproven for the two-dimensional Kelvin-Helmholtzproblem where a minimal regularity implies the realanalyticity of the interface. Eventually, we usethis shear flow to provide explicit examples ofnon-regular solutions of the three-dimensionalEuler equations that conserve the energy, an issuewhich is related to the Onsager conjecture.

This is a joint work with Claude Bardos.
Mon, 07/02/2011
17:00 		Grigoris Pavliotis (Imperial College) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Asymptotic analysis for the Generalized Langevin equation
In this talk we will present some recent results on the long time asymptotics of the generalized (non-Markovian) Langevin equation (gLE). In particular, we will discuss about the ergodic properties of the gLE and present estimates on the rate of convergence to equilibrium, we will present a homogenization result (invariance principle) and we will discuss about the convergence of the gLE dynamics to the (Markovian) Langevin dynamics, in some appropriate asymptotic limit. The analysis is based on the approximation of the gLE by a high (and possibly infinite) dimensional degenerate Markovian system, and on the analysis of the spectrum of the generator of this Markov process. This is joint work with M. Ottobre and K. Pravda-Starov.
Mon, 14/02/2011
17:00 		James Robinson (University of Warwick) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Numerical verification of regularity for solutions of the 3D Navier-Stokes equations
I will show that one can (at least in theory) guarantee the "validity" of a numerical approximation of a solution of the 3D Navier-Stokes equations using an explicit a posteriori test, despite the fact that the existence of a unique solution is not known for arbitrary initial data. The argument relies on the fact that if a regular solution exists for some given initial condition, a regular solution also exists for nearby initial data ("robustness of regularity"); I will outline the proof of robustness of regularity for initial data in $ H^{1/2} $. I will also show how this can be used to prove that one can verify numerically (at least in theory) the following statement, for any fixed R > 0: every initial condition $ u_0\in H^1 $ with $ \|u\|_{H^1}\le R $ gives rise to a solution of the unforced equation that remains regular for all $ t\ge 0 $. This is based on joint work with Sergei Chernysehnko (Imperial), Peter Constantin (Chicago), Masoumeh Dashti (Warwick), Pedro Marín-Rubio (Seville), Witold Sadowski (Warsaw/Warwick), and Edriss Titi (UC Irivine/Weizmann).
Mon, 21/02/2011
17:00 		Marco Cicalese (Universita die Napoli) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
The isoperimetric inequality in quantitative form
The classical isoperimetric inequality states that, given a set $ E $ in $ R^n $ having the same measure of the unit ball $ B $, the perimeter $ P(E) $ of $ E $ is greater than the perimeter $ P(B) $ of $ B $. Moreover, when the isoperimetric deficit $ D(E)=P(E)-P(B) $ equals 0, than $ E $ coincides (up to a translation) with $ B $. The sharp quantitative form of the isoperimetric inequality states that $ D(E) $ can be bound from below by $ A(E)^2 $, where the Fraenkel asymmetry $ A(E) $ of $ E $ is defined as the minimum of the volume of the symmetric difference between $ E $ and any translation of $ B $. This result, conjectured by Hall in 1990, has been proven in its full generality by Fusco-Maggi-Pratelli (Ann. of Math. 2008) via symmetrization arguments and more recently by Figalli-Maggi-Pratelli (Invent. Math. 2010) through optimal transportation techniques. In this talk I will present a new proof of the sharp quantitative version of the isoperimetric inequality that I have recently obtained in collaboration with G.P.Leonardi (University of Modena e Reggio). The proof relies on a variational method in which a penalization technique is combined with the regularity theory for quasiminimizers of the perimeter. As a further application of this method I will present a positive answer to another conjecture posed by Hall in 1992 concerning the best constant for the quantitative isoperimetric inequality in $ R^2 $ in the small asymmetry regime.
Mon, 28/02/2011
17:00 		Matthias Röger (Technische Universität Dormund) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Stochastic perturbations of the Allen-Cahn equation
In this talk we will first consider the Allen-Cahn action functional that controls the probability of rare events in an Allen-Cahn type equation with additive noise. Further we discuss a perturbation of the Allen-Cahn equation by a stochastic flow. Here we will present a tightness result in the sharp interface limit and discuss the relation to a version of stochastically perturbed mean curvature flow. (This is joint work with Luca Mugnai, Leipzig, and Hendrik Weber, Warwick.)
Mon, 07/03/2011
17:00 		Filip Rindler (University of Oxford) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Lower semicontinuity in the space BD of functions of bounded deformation
The space BD of functions of bounded deformation consists of all L1-functions whose distributional symmetrized derivative (defined by duality with the symmetrized gradient ($ \nabla u + \nabla u^T)/2 $) is representable as a finite Radon measure. Such functions play an important role in a variety of variational models involving (linear) elasto-plasticity. In this talk, I will present the first general lower semicontinuity theorem for symmetric-quasiconvex integral functionals with linear growth on the whole space BD. In particular we allow for non-vanishing Cantor-parts in the symmetrized derivative, which correspond to fractal phenomena. The proof is accomplished via Jensen-type inequalities for generalized Young measures and a construction of good blow-ups, based on local rigidity arguments for some differential inclusions involving symmetrized gradients. This strategy allows us to establish the lower semicontinuity result even without a BD-analogue of Alberti's Rank-One Theorem in BV, which is not available at present. A similar strategy also makes it possible to give a proof of the classical lower semicontinuity theorem in BV without invoking Alberti's Theorem.

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

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