Abstract
Stochastic 2D Navier–Stokes with maximally degenerate noise and its Galerkin approximations
We study the stochastic Navier–Stokes equations on a 2D torus with the noise degenerated to a single Fourier mode.
The aim is to complement the results by Hairer and Mattingly on the uniqueness of the invariant measure for the Navier–Stokes equations.
When the noise acts on the lowest Fourier mode, we prove the uniqueness of the invariant measure.
We then characterise all possible 3-dimensional Galerkin approximations of the Navier–Stokes equations. For a very similar 3-dimensional system we show the existence of nontrivial invariant measures, a result similar to Coti Zelati and Hairer.
More precisely, we prove that a nontrivial invariant measure exists if and only if the corresponding deterministic system admits a nontrivial fixed point.
The presented results were obtained jointly with Prof. Szymon Peszat.