Ergodicity of a Galerkin approximation of three-dimensional magnetohydrodynamics system forced by a degenerate noise |
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Authors: | Kazuo Yamazaki |
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Institution: | Department of Mathematics, University of Rochester, Rochester, NY, USA |
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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 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. |
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Keywords: | Ergodicity Harris' condition Hörmander's condition hypoellipticity invariant measure magnetohydrodynamics system |
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