Transport equations and fluid mechanics.

Summary The object of this thesis is the mathematical analysis of models from fluid mechanics. The study is mainly focused on the incompressible inhomogeneous Navier-Stokes equations and the compressible isentropic Navier-Stokes equations. The first part is devoted to ordinary differential equations associated with vector fields with irregular coefficients, typically with integrable derivatives. R. J. Di Perna and P. -L. Lions have been pioneers in the study of vector fields with W#1#,#1 regularity and bounded divergence, by showing the existence and uniqueness of a flow X verifying most of the properties of regular vector field flows, valid however for almost any initial point. The object of the first part is to extend this theory to fields with unbounded divergence. The proof is based on the method of normalized solutions for transport equations, introduced by R. J. Di Perna and P. -L. Lions. In the continuity of the previous results, we show on the other hand an existence theorem of stronger solutions corresponding to initial data in W#1#,#m (m > 1) for #t +b. * = 0, the associated vector field b being assumed to have Sobolev regularity W#s#+#1#,#p with sp = n. These results are then applied to a proof of uniqueness of solutions of the incompressible inhomogeneous Navier-Stokes equations in dimension 2. In the second part of this work, we focus on incompressible fluid models. We consider a family of incompressible immiscible fluids indexed by 1, . m in an open r#n (n 2). These fluids are characterized by their density i#1im and their viscosity #i#1##im. The first chapter deals with the global existence of weak solutions for the incompressible Navier-Stokes equations when the domain is unbounded. The regularity of multiphase plane flows is then studied, stating the results in terms of the relative dispersion of viscosities, while taking into account the possible presence of vacuum pockets in the fluid medium. The third chapter is devoted to some remarks on the regularity of the weak solutions of an equation from a simplified model of magnetohydrodynamics, coupling the incompressible Navier-Stokes equations and the Maxwell equations. Finally, we study the Navier-Stokes equations modeling the evolution of an isentropic compressible fluid. The work of P.-L. Lions ensures the global existence in time of weak solutions under certain assumptions on the pressure law. In dimension n = 2 or 3, one can show results of regularity in small time for initial densities canceling. When n = 2, one obtains global results in time, provided that the density remains bounded. We use a logarithmic estimate, demonstrated in the context of the incompressible models mentioned above. In the second chapter, we analyze the regularity of weak solutions in dimension n 2, by showing an a priori estimate which gives information on the regularity in time of the velocity field.