Forthcoming Seminars › Seminar Series

Partial Differential Equations Seminar

Mon, 21/04/2008
17:00 		V.P. Smyshlyaev (University of Bath) Partial Differential Equations Seminar Add to calendar L3
Multi-phase mixtures, multi-well relaxation and $ H $-measures
Multi-well relaxation problem emerges e.g. in characterising effective properties of composites and in phase transformations. This is a nonlinear problem and one approach uses its reformulation in Fourier space, known in the theory of composites as Hashin-Shtrikman approach, adapted to nonlinear composites by Talbot and Willis. Characterisation of admissible mixtures, subjected to appropriate differential constraints, leads to a quasiconvexification problem. The latter is equivalently reformulated in the Fourier space as minimisation with respect to (extremal points of) H-measures of characteristic functions (Kohn), which in a sense separates the microgeometry of mixing from the differential constraints. For three-phase mixtures in 3D we obtain a full characterisation of certain extremal H-measures. This employs Muller's Haar wavelet expansion estimates in terms of Riesz transform to establish via the tools of harmonic analysis weak lower semicontinuity of certain functionals with rank-2 convex integrands. As a by-product, this allows to fully solve the problem of characterisation of quasiconvex hulls for three arbitrary divergence-free wells. We discuss the applicability of the results to problems with other kinematic constraints, and other generalisations. Joint work with Mariapia Palombaro, Leipzig.
Mon, 28/04/2008
17:00 		H. Beirao da Veiga (Pisa) Partial Differential Equations Seminar Add to calendar L3
Concerning the Ladyzhenskaya-Smagorinsky turbulence model - The regularity problem
we present some sharp regularity results for the stationary and the evolution Navier-Stokes equations with shear dependent viscosity, under the no-slip boundary condition. This is a classical turbulence model, considered by von Neumann and Richtmeyer in the 50's, and by Smagorinski in the beginning of the 60's (for p= 3). The model was extended to other physical situations, and deeply studied from a mathematical point of view, by Ladyzhenskaya in the second half of the 60's. We consider the shear thickening case p>2. We are interested in regularity results in Sobolev spaces, up to the boundary, in dimension n=3, for the second order derivatives of the velocity and the first order derivatives of the pressure. In spite of the very rich literature on the subject, sharp regularity results up to the boundary are quite new.
Mon, 05/05/2008
17:00 		J. Conlon (University of Michigan, USA) Partial Differential Equations Seminar Add to calendar L3
The Becker-Doering (B-D) and Lifschitz-Slyozov-Wagner (LSW) Equations
The B-D equations describe a mean field approximation for a many body system in relaxation to equilibrium. The two B-D equations determine the time evolution of the density c(L,t) of particles with mass L, L=1,2,... One of the equations is a discretized linear diffusion equation for c(L,t), and the other is a non-local constraint equivalent to mass conservation. Existence and uniqueness for the B-D system was established in the 1980's by Ball, Carr and Penrose. Research in the past decade has concentrated on understanding the large time behavior of solutions to the B-D system. This behavior is characterized by the phenomenon of "coarsening", whereby excess density is concentrated in large particles with mass increasing at a definite rate. An important conjecture in the field is that the coarsening rate can be obtained from a particular self- similar solution of the simpler LSW system. In this talk we shall discuss the B-D and LSW equations, and some recent progress by the speaker and others towards the resolution of this conjecture.
Mon, 12/05/2008
17:00 		Elise Fouassier (Université de Toulouse) Partial Differential Equations Seminar Add to calendar L3
High frequency limit of Helmholtz equations : refraction by sharp interfaces
We compute the high frequency limit of the Hemholtz equation with source term, in the case of a refraction index that is discontinuous along a sharp interface between two unbounded media. The asymptotic propagation of energy is studied using Wigner measures. First, in the general case, assuming some geometrical hypotheses on the index and assuming that the interface does not capture energy asymptotically, we prove that the limiting Wigner measure satisfies a stationary transport equation with source term. This result encodes the refraction phenomenon. Second, we study the particular case when the index is constant in each media, for which the analysis goes further: we prove that the interface does not capture energy asymptotically in this case.
Mon, 26/05/2008
17:00 		Miguel Escobedo (Universidad del País Vasco) Partial Differential Equations Seminar Add to calendar L3
Singular solutions for homogeneous quantum Boltzmann equations
Mon, 09/06/2008
17:00 		James Robinson (Warwick) Partial Differential Equations Seminar Add to calendar L3
Uniqueness of Lagrangian trajectories for weak solutions of the two- and three-dimensional Navier-Stokes equations
I will discuss recent results concerning the uniqueness of Lagrangian particle trajectories associated to weak solutions of the Navier-Stokes equations. In two dimensions, for which the weak solutions are unique, I will present a mcuh simpler argument than that of Chemin & Lerner that guarantees the uniqueness of these trajectories (this is joint work with Masoumeh Dashti, Warwick). In three dimensions, given a particular weak solution, Foias, Guillopé, & Temam showed that one can construct at leaset one trajectory mapping that respects the volume-preserving nature of the underlying flow. I will show that under the additional assumption that $ u\in L^{6/5}(0,T;L^\infty) $ this trajectory mapping is in fact unique (joint work with Witek Sadowski, Warsaw).

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