Asymptotic Analysis

In a previous paper the second author studied the modelling of piezoelectric static thin plates. The present work is concerned with the dynamical case. Magnetic effects are taken into account and we deal with the complete system of Maxwell equations. Moreover, we give an analysis for both an insulated plate and a plate submitted to a difference of potential. These two cases give rise to different behaviours.

We consider a weakly coupled semilinear parabolic‐hyperbolic system with a degenerate and anisotropic diffusion. It arises to model the evolution of a chemical or biological tracer in a porous medium. We study the well‐posedness of the system using a L¹ theory. Then, we establish the relaxation limit as the reaction constant becomes large. We prove that the system converges to a nonlinear parabolic‐hyperbolic equation that generalizes the Stefan problem. Two specificities of this paper are (i) to deal with ill‐prepared initial data and (ii) with unique entropy solutions based on a precise entropy inequality.

We study nonlinear equations subject to oscillatory initial data. The oscillatory solution of such problems tends to a homogenized weak limit that is characterized by the corresponding homogenized equations. Those equations usually involve an additional independent variable, so that the weak limit is an average of infinitely many functions. In certain cases, however, there is an alternative description to the weak limit via a closed finite system of equations that the weak limit and some of its moments satisfy. We study the question of an existence of such finite closures in the context of semilinear Boltzmann type equations and the quasilinear Euler equations and show that, in most cases, finite closures do not exist.

In this paper we study the existence and asymptotic behavior of the global solutions of some degenerate parabolic equation with critical Sobolev exponent. In particular, we apply the concentration‐compactness principle of P.‐L. Lions to the study of the asymptotic behavior of global solutions with the initial value in “stable set”.