Ergodicity of a Galerkin approximation of three-dimensional magnetohydrodynamics system forced by a degenerate noise

TL;DR

This paper proves the existence and uniqueness of an ergodic invariant measure for a Galerkin approximation of the 3D magnetohydrodynamics system with degenerate noise, extending ergodic theory to complex coupled PDEs.

Contribution

It establishes ergodicity and invariant measure existence for a Galerkin approximation of 3D MHD equations, incorporating complex nonlinear coupling.

Findings

Existence of a unique invariant measure for the Galerkin system.

Proof of ergodicity for the coupled MHD system.

Handling of nonlinear coupling complexities in the proof.

Abstract

Magnetohydrodynamics system consists of a coupling of the Navier-Stokes and Maxwell's equations and is most useful in studying the motion of electrically conducting fluids. We prove the existence of a unique invariant, and consequently ergodic, measure for the Galerkin approximation system of the three-dimensional magnetohydrodynamics system. The proof is inspired by those of \cite{EM01, R04} on the Navier-Stokes equations; however, computations involve significantly more complications due to the coupling of the velocity field equations with those of magnetic field that consists of four non-linear terms.