Past Partial Differential Equations Seminar

12 May 2008
17:00
Elise Fouassier
Abstract
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.
  • Partial Differential Equations Seminar
5 May 2008
17:00
Abstract
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.
  • Partial Differential Equations Seminar
28 April 2008
17:00
H. Beirao da Veiga
Abstract
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.
  • Partial Differential Equations Seminar
21 April 2008
17:00
V.P. Smyshlyaev
Abstract
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) <i>H</i>-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 <i>H</i>-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.
  • Partial Differential Equations Seminar
3 March 2008
16:00
Sorin Mardare
Abstract
One of the intrinsic methods in elasticity is to consider the Cauchy-Green tensor as the primary unknown, instead of the deformation realizing this tensor, as in the classical approach. Then one can ask whether it is possible to recover the deformation from its Cauchy-Green tensor. From a differential geometry viewpoint, this amounts to finding an isometric immersion of a Riemannian manifold into the Euclidian space of the same dimension, say d. It is well known that this is possible, at least locally, if and only if the Riemann curvature tensor vanishes. However, the classical results assume at least a C2 regularity for the Cauchy-Green tensor (a.k.a. the metric tensor). From an elasticity theory viewpoint, weaker regularity assumptions on the data would be suitable. We generalize this classical result under the hypothesis that the Cauchy-Green tensor is only of class W^{1,p} for some p>d. The proof is based on a general result of PDE concerning the solvability and stability of a system of first order partial differential equations with L^p coefficients.
  • Partial Differential Equations Seminar
25 February 2008
16:00
Abstract
We discuss a model of finite strain gradient plasticity including phenomenological Prager type linear kinematical hardening and nonlocal kinematical hardening due to dislocation interaction. Based on the multiplicative decomposition a thermodynamically admissible flow rule for <i>F<sub>p</sub></i> is described involving as plastic gradient Curl <i>F<sub>p</sub></i>. The formulation is covariant w.r.t. superposed rigid rotations of the reference, intermediate and spatial configuration but the model is not spin-free due to the nonlocal dislocation interaction and cannot be reduced to a dependenceon the plastic metric <i>C<sub>p</sub>=F<sub>p</sub><sup>T</sup> F<sub>p</sub></i>. <br> The linearization leads to a thermodynamically admissible model of infinitesimal plasticity involving only the Curl of the non-symmetric plastic distortion <i>p</i>. Linearized spatial and material covariance under constant infinitesimal rotations is satisfied. <br> Uniqueness of strong solutions of the infinitesimal model is obtained if two non-classical boundary conditions on the plastic distortion <i>p</i> are introduced: d<sub><i>t</i></sub><i>p</i>.&tau;=0 on the microscopically hard boundary &Gamma;<sub><i>D</i></sub>&sub;&part;&Omega; and [Curl<i>p</i>].&tau;=<i>0</i> on the microscopically free boundary &part;&Omega;\&Gamma;<sub><i>D</i></sub>, where &tau; are the tangential vectors at the boundary &part;&Omega;. Moreover, I show that a weak reformulation of the infinitesimal model allows for a global in-time solution of the corresponding rate-independent initial boundary value problem. The method of choice are a formulation as a quasivariational inequality with symmetric and coercive bilinear form, following the abstract framework proposed by Reddy. Use is made of new Hilbert-space suitable for dislocation density dependent plasticity.
  • Partial Differential Equations Seminar

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