Abstract.We study the stationary flow of a fluid occupying a 3D infinite horizontal domain bounded by a rough wall that is at rest and by a plane that moves with a constant velocity. The rough wall is a plane covered with periodically distributed asperities of size e. We prove that, outside a neighbourhood of the rough region, the flow behaves asymptotically as a Couette flow, as e -» 0, up to an exponentially small error.

We study the differential system governing the flow of an incompressible ferrofluid under the action of a magnetic field. The system consists of the Navier-Stokes equations, the angular momentum equation, the magnetization equation, and the magnetostatic equations. We prove, by using the Galerkin method, a global in time existence of weak solutions with finite energy of an initial boundary-value problem and establish the long-time behavior of such solutions. The main difficulty is due to the singularity of the gradient magnetic force. (2000). 35Q35, 76D05. Mathematics Subject Classification

We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics. RésuméNousétudions le comportement asymptotique des solutions d'un problème spectral associéà l'opérateur de Laplace dans un domaineà frontière oscillante. Nous considérons le cas où la valeur propre du problème limite est multiple. Nous construisons les termes principaux des développements asymptotiques deséléments propres et nous donnons une justification rigoureuse des approximations asymptotiques.