Global regularity for 3D magneto-hydrodynamics equations with only horizontal dissipation

Abstract

We investigate the Cauchy problem for the 3D magnetohydrodynamics equations with only horizontal dissipation for the small initial data. With the help of the dissipation in the horizontal direction and the structure of the system, we analyze the properties of the decay of the solution and apply these decay properties to get the global regularity of the solution. In the process, we mainly use the frequency decomposition in Green's function method and energy method.

Key words： Magneto-hydrodynamics (MHD) equations; degenerate dissipation; global solution; Green's function method; frequency decomposition method

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