Symétrie et compacité dans les espaces de Sobolev

In this paper, we show several compactness results concerning the subspaces of general Sobolev spaces formed by the functions possessing some symmetry: spherical symmetry, cylindrical symmetry, etc.     

Inéquations en thermoélasticité et magnétohydrodynamique

Sans résumé     

Renormalized solutions of some transport equations with partially <Emphasis Type="Italic">W</Emphasis><Superscript>1,1</Superscript> velocities and applications

We prove existence and uniqueness of renormalized solutions of some transport equations with a vector field that is not W^1,1 with respect to all variables but is of a particular form. Two specific applications of this new result are then treated, based upon the equivalence between transport equations and ordinary differential equations. The first one consists of a result about the dependance upon initial conditions for solutions of ODEs. The second one is related to some stochastic differential equations arising in the modelling of polymeric fluid flows.     

Contribution à un problème de M. M. Picone

Résumé Par une méthode fonctionnelle développée ailleurs (cf. la bibliographie), on apporte dans cet article une contribution à un problème posé parM. M. Picone. On montre que, sur des ouverts dont la frontière est suffisamment régulière, le problème deM. M. Picone (et même un type plus général de problèmes aux limites) admet une solution unique, les conditions aux limites étant prises ? en moyenne ? (cf. N^o 6 pour les définitions précises).     

SENTINELS FOR PERIODIC DISTRIBUTED SYSTEMS

For periodic distributed systems the author reduces the "sentinels" problem to a problem of controllability type and uses suitable adaptations of HUM (Hilbert Uniqueness Method) to gives solutions to the original “sentinels” problem.     

A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description

We consider the canonical Gibbs measure associated to aN-vortex system in a bounded domain Λ, at inverse temperature \(\widetilde\beta \) and prove that, in the limitN→∞, \(\widetilde\beta \) /N→β, αN→1, where β∈(?8π, + ∞) (here α denotes the vorticity intensity of each vortex), the one particle distribution function ?^N = ? ^N x,x∈Λ converges to a superposition of solutions ? _α of the following Mean Field Equation: $$\left\{ {\begin{array}{*{20}c} {\varrho _{\beta (x) = } \frac{{e^{ - \beta \psi } }}{{\mathop \smallint \limits_\Lambda e^{ - \beta \psi } }}; - \Delta \psi = \varrho _\beta in\Lambda } \\ {\psi |_{\partial \Lambda } = 0.} \\ \end{array} } \right.$$ Moreover, we study the variational principles associated to Eq. (A.1) and prove thai, when β→?8π⁺, either ?_β → δ _{x 0} (weakly in the sense of measures) wherex ₀ denotes and equilibrium point of a single point vortex in Λ, or ?_β converges to a smooth solution of (A.1) for β=?8π. Examples of both possibilities are given, although we are not able to solve the alternative for a given Λ. Finally, we discuss a possible connection of the present analysis with the 2-D turbulence.     

Boundary regularity for some nonlinear elliptic degenerate equations

We consider the nonlinear elliptic degenerate equation (1) $$ - x^2 \left( {\frac{{\partial ^2 u}}{{\partial x^2 }} + \frac{{\partial ^2 u}}{{\partial y^2 }}} \right) + 2u = f(u)in\Omega _a ,$$ where $$\Omega _a = \left\{ {(x,y) \in \mathbb{R}^2 ,0< x< a,\left| y \right|< a} \right\}$$ for some constanta>0 andf is aC ^∞ functions on ? such thatf(0)=f′(0)=0. Our main result asserts that: ifu∈C \((\bar \Omega _a )\) satisfies (2) $$u(0,y) = 0for\left| y \right|< a,$$ thenx ^?2 u(x,y)∈C ^∞ \(\left( {\bar \Omega _{a/2} } \right)\) and in particularu∈C ^∞ \(\left( {\bar \Omega _{a/2} } \right)\) .     

Mean-field games with a major player

We introduce and study mathematically a new class of mean-field-game systems of equations. This class of equations allows us to model situations involving one major player (or agent) and a “large” group of “small” players.