TY - JOUR
T1 - Ergodicity of a Galerkin approximation of three-dimensional magnetohydrodynamics system forced by a degenerate noise
AU - Yamazaki, Kazuo
N1 - Publisher Copyright:
© 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group.
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
SN - 1744-2508
VL - 91
SP - 114
EP - 142
JO - Stochastics
JF - Stochastics
IS - 1
ER -