On SDEs with Partial Damping Inspired by the Navier-Stokes Equations

Abstract

The solution to the Navier Stokes equations on the 2D torus with stochastic forcing that is white noise in time, coloured in space has a Fourier series representation whose coordinates satisfy a countable system of Stochastic Differential Equations. Inspired by the structure of these equations, we construct a finite system of stochastic differential equations with a similar structure and explore the conditions under which the system has an invariant distribution. \\Our main tool to prove existence of invariant distributions are Lyapunov functions, or more generally Lyapunov pairs. In particular, we construct the Lyapunov pairs piecewise over different regions and then use mollifiers to unify these disparate characterisations. We also apply some results from Algebraic Geometry and Matrix Perturbation Theory to study and exploit the geometry of the problem in high dimensions. \\The combination of these methods allowed us to prove that a large class of the equations we constructed have an invariant distributions. Furthermore we have explicit tail estimates for these invariant distributions.