An Introduction to Stochastic Navier–Stokes Equations

Abstract

The dynamics of liquids and gases can be modeled by the Navier–Stokes system of partial differential equations describing the balance of mass and momentum in the fluid flow. In recent years their has been an increasing interest in random influences on the fluid motion modeled via stochastic partial differential equations.

In this lecture notes we study the existence of weak martingale solutions to the stochastic Navier-Stokes equations (both incompressible and compressible). These solutions are weak in the analytical sense (derivatives exists only in the sense of distributions) and weak in the stochastic sense (the underlying probability space is not a priori given but part of the problem). In particular, we give a detailed introduction to the stochastic compactness method based on Skorokhod’s representation theorem.