TY - JOUR

T1 - Ergodicity of a Galerkin approximation of three-dimensional magnetohydrodynamics system forced by a degenerate noise

AU - Yamazaki, Kazuo

PY - 2019/1/2

Y1 - 2019/1/2

N2 - 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 [E. Weinan and J.C. Mattingly, Ergodicity for the Navier-Stokes equation with degenerate random forcing: Finitedimensional approximation, Comm. Pure Appl. Math. LIV (2001), pp. 1386–1402; M. Romito, Ergodicity of the finite dimensional approximation of the 3D Navier-Stokes equations forced by a degenerate noise, J. Stat. Phys. 114 (2004), pp. 155–177] 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.

AB - 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 [E. Weinan and J.C. Mattingly, Ergodicity for the Navier-Stokes equation with degenerate random forcing: Finitedimensional approximation, Comm. Pure Appl. Math. LIV (2001), pp. 1386–1402; M. Romito, Ergodicity of the finite dimensional approximation of the 3D Navier-Stokes equations forced by a degenerate noise, J. Stat. Phys. 114 (2004), pp. 155–177] 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.

KW - 35Q35

KW - 37L55

KW - 60H15

KW - Ergodicity

KW - Harris' condition

KW - Hörmander's condition

KW - hypoellipticity

KW - invariant measure

KW - magnetohydrodynamics system

UR - http://www.scopus.com/inward/record.url?scp=85053303712&partnerID=8YFLogxK

U2 - 10.1080/17442508.2018.1518984

DO - 10.1080/17442508.2018.1518984

M3 - Article

AN - SCOPUS:85053303712

VL - 91

SP - 114

EP - 142

JO - Stochastics

JF - Stochastics

SN - 1744-2508

IS - 1

ER -