Séminaire Analyse non linéaire et EDP

We prove a blow-up criterion in terms of the upper bound ofthe density for the strong solution to the 3-D compressible Navier-Stokesequations. The initial vacuum is allowed. The main ingredient of theproof is a priori estimate for an important quantity under the assumptionthat the density is upper bounded, whose divergence can be viewed asthe effective viscous flux.

Some twenty years ago Berenger introduced theremarkable method of perfectly matchedlayers for truncating to a rectangle, the computation ofsolutions of Maxwell's equations in 1+2 and 1+3 dimensionalspace time. Only recently have some of the fundamentalquestions concerning this method been resolved.For example the stability of the original methodand its perfection. We discuss the analysis of thisand related methods that are constructed to performbetter in variable coefficient settings where the perfectionof Berenger no longer holds. Research donewith Laurence Haplern, Sabrina Petit, and LudovicMetivier.

On considère le système de Zakharov dans R3. Ce dernier décrit la propagationdes ondes de Langmuir dans un plasma faiblement magnétisé. Desarguments heuristiques et des simulations numériques ont montré que les solutionspeuvent devenir singulières au bout d?RTMun temps fini pour des donnéesinitiales assez ?R~grandes?RTM.Dans ce travail, on suppose que la solution explose en temps fini et onétablit une bonne inférieure pour le taux d?RTMexplosion de certaines normes deSobolev de la solution. L?RTManalyse est basée sur la théorie d?RTMexistence localede Ginibre-Tsutsumi-Velo (1997) et un argument de contradiction développépar Cazenave-Weissler (1990) dans […]

We discuss relations between one-dimensional inviscid and viscous stability/bifurcation of shock waves in continuum-mechanical systems and existence of a convex entropy. In particular, we show that the equations of gas dynamics admit equations of state satisfying all of the usual assumptions of an ideal gas, along with thermodynamic stability- i.e., existence of a convex entropy- yet for which there occur unstable inviscid shock waves. For general 3í3 systems (but not up to now gas dynamics), we give numerical evidence showing that viscous shocks can exhibit Hopf bifurcation to pulsating shock […]