Soluciones débiles en mecánica de fluidos

Abstract

The main aim of this work is to prove theoretical results on partial differential equations from fluid mechanics. Particularly, the theoretical development is destined to prove the existence of weak solutions of the Navier-Stokes equations in two and three dimensions. The Navier-Stokes equations is a classical topic in the study of the dynamics of incompressible viscous fluids. Those equations present basic and important open questions such as regularity and finite time singularity formation of the solutions. It is a current area of mathematical research of fundamental interest in particular due to its physical relevance and broad applicability. The first chapter introduces concepts of Functional Analysis that go beyond the scope of what is taught in the degree, and will be very useful in the development of this work. It establishes the main concepts and results related to weak convergence, needed to understand the concept of weak solution. The goal of the first section is to prove the weak compactness of bounded sets in a Hilbert space. Next, we take on the evolution problem of second-order parabolic equations, which has as a representative example the Heat equation. We use variational formulation to prove the existence of weak solutions. For that, we analyze the properties of the terms of the equation. On these properties we will prove some results of continuity and compactness, and we will finally apply Galerkin method. Due to the good properties of the equation, the results are proven in an arbitrary finite dimension and the uniqueness of the solution is proven as well. The techniques used for the study of the equation are repeated in the Navier-Stokes case. This first chapter also serves for acquiring familiarity with the method. The second chapter deals with the Navier-Stokes equation in the complete space of two and three dimensions with the same techniques as in the previous chapter. We will find greater difficulties due mainly to non-linearity. It begins by introducing the usual Hilbert spaces of fluid problems that incorporate incompressibility, and provides results that allow us to tackle the pressure of the equation, simplifying the problem. Next, we analyze in detail the non-linear term, finding a limitation in the dimension of the workspace. After introducing the variational formulation, compactness theorems which are necessary to treat the non-linear term are proven. Finally, the Galerkin method is applied again, and the existence of weak solutions in the cases of two and three dimensions is proved. The uniqueness in the two-dimensional case is also tested. This problem was originally studied by Jean Leray, who proved in 1934 the existence of weak solutions. For the three-dimensional case, it has recently been proven that there is no uniqueness of weak solutions.