Gibbsian dynamics and ergodicity of magnetohydrodynamics and related systems forced by random noise

Abstract

The magnetohydrodynamics system consists of the Navier-Stokes equations from fluid mechanics, coupled with the Maxwell’s equations from electromagnetism through multiples of non-linear terms involving derivatives. Following the approach of [1], we prove the existence of a unique invariant measure in case the forcing terms consist of the cylindrical Wiener processes with only low modes. Its proof requires taking advantage of the structure of the non-linear terms carefully and is extended to various other related models such as the magnetohydrodynamics-Boussinesq system from fluid mechanics in atmosphere and oceans, as well as the magneto-micropolar fluid system from the theory of microfluids.

Keywords:

• Boussinesq system
• ergodicity
• magnetohydrodynamics
• micropolar fluid
• Navier-Stokes equations

MATHEMATICS SUBJECT CLASSIFICATION: 2010:

• MSC: 35Q35
• 37L55
• 60H15

Acknowledgments

The author expresses deep gratitude to the Editor and the anonymous referee for valuable comments and suggestions which improved this manuscript significantly.