Les eqüacions de Navier-Stokes : existència de solucions febles

Abstract

In this review, we introduce the first step in order to study the Navier-Stokes' system. We focus our attention in the steady, three dimensional case. First of all, we begin by introducing the equations system with the motivation of the physical problem of a viscous fluid. In a first approach, we consider only transversal to the surface of a fluid region forces, in order to introduce later another non-transversal term. Then, we define the smooth functions spaces, Lebesgue spaces and we introduce the theory of distributions. In this section, some important inequalities are presented, such as H ̈older’s inequality. Also, Sobolev spaces are defined. Finally, the Hahn-Banach Theorem and Riesz Theorem are proven. In the last section, we study the Navier-Stokes' system from a distribution point of view and the concept of a weak solution is defined. After that, some results in the non-linear term are presented and we give a way to construct the pressure associated to a given weak solution. Finally, two results concerning existence of weak solutions are proven, one in bounded domains and the other one in non bounded ones. Also, we obtain the suficient conditions in the external force: $f\in W^{-1,2} (\Omega) ^{3}$ and $f\in L^{6/5} (\Omega) ^{3}$