Smoothness of the trajectories of ideal fluid particles with Yudovich vorticities in a planar bounded domain

We consider the incompressible Euler equations in a (possibly multiply connected) bounded domain of R^2, for flows with bounded vorticity, for which Yudovich proved, in 1963, global existence and uniqueness of the solution. We prove that if the boundary of the domain is C^infty (respectively Gevrey of order M > 1) then the trajectories of the fluid particles are C^infty (resp. Gevrey of order M + 2). Our results also cover the case of "slightly unbounded" vorticities for which Yudovich extended his analysis in 1995. Moreover if in addition the initial vorticity is Holder continuous on a part of the domain then this Holder regularity propagates smoothly along the flow lines. Finally we observe that if the vorticity is constant in a neighborhood of the boundary, the smoothness of the boundary is not necessary for these results to hold

A Report on Six Seminars About the UK Supreme Court

During the first half of 2008, a series of six seminars was held in the School of Law at Queen Mary University of London on the United Kingdom Supreme Court. Participants included Law Lords and other senior members of the judiciary, practitioners, and academics. This report records the fascinating exchange of views that took place at those unprecedented meetings. Among the themes explored were: the selection of cases; relations of the UK Supreme Court with lower courts and tribunals; procedures and costs; communication methods; the UK Supreme Court's jurisdiction over Scottish matters; and the constitutional framework within which the new court will work

On the weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-Magneto-Hydrodynamic equations

In this paper we deal with weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-Magneto-Hydrodynamic equations. First we prove that these solutions satisfy some weak-strong uniqueness property. Then we investigate the validity of energy identities. In particular we give a sufficient condition on the regularity of weak solutions to rule out anomalous dissipation. In the case of the Hall-Magneto-Hydrodynamic equations we also give a sufficient condition to guarantee the magneto-helicity identity. Our conditions correspond to the same heuristic scaling as the one introduced by Onsager in hydrodynamic theory. Finally we examine the sign, locally, of the anomalous dissipations of weak solutions obtained by some natural approximation processes.Comment: 45 page

Cauchy problem and quasi-stationary limit for the Maxwell-Landau-Lifschitz and Maxwell-Bloch equations

In this paper we continue the investigation of the Maxwell-Landau-Lifschitz and Maxwell-Bloch equations. In particular we extend some previous results about the Cauchy problem and the quasi-stationary limit to the case where the magnetic permeability and the electric permittivity are variable

On the ferromagnetism equations with large variations solutions

We exhibit some large variations solutions of the Landau-Lifschitz equations as the exchange coefficient ε^2 tends to zero. These solutions are described by some asymptotic expansions which involve some internals layers by means of some large amplitude fluctuations in a neighborhood of width of order ε of an hypersurface contained in the domain. Despite the nonlinear behaviour of these layers we manage to justify locally in time these asymptotic expansions