Navier-Stokes equations and forward-backward SDEs on the group of diffeomorphisms of a torus

Abstract

Throughout this study, our general objective was to establish a relationship between the solutions of a forward-backward system of stochastic differential equations in the group of diffeomorphisms of the flat torus that are Hα-smooth for α ≥ 2 which preserve volume (denoted as Dα v (T2)) and the solutions of the Navier-Stokes equations in the plane, R2. The theory in Chapters 1 and 2 was obtained mainly by Shkoller [16], do Carmo [6] and Gliklikh [10] to provide the basic concepts about Sobolev spaces, Riemannian Geometry and Stochastic Analysis on Manifolds that would be necessary for the development of the results of Cruzeiro and Shamarova [4]. In Chapter 3, we present the first main result of the work. We assume that there exists a solution to the Navier-Stokes equations in the plane, and from this solution we find a triple of solutions to a system of forward-backward stochastic differential equations in Dα v (T2). Finally, in Chapter 4, we present the converse result and given a solution of the aforementioned system, we construct a solution to Navier-Stokes equations.