Abstract
We study a regularised version of the magnetohydrodynamics (MHD) equations, the tamed MHD (TMHD) equations. They are a model for the flow of electrically conducting fluids through porous media. We prove existence and uniqueness of TMHD on the whole space {mathbb {R}}^{3}, that smooth data give rise to smooth solutions and show that solutions to TMHD converge to a suitable weak solution of the MHD equations as the taming parameter N tends to infinity. Furthermore, we adapt a regularity result for the Navier–Stokes equations to the MHD case.
Highlights
The magnetohydrodynamics (MHD) equations describe the dynamic motion of electrically conducting fluids
Before we study the tamed equations, we want to give an overview of regularisation schemes for the Navier–Stokes and the MHD equations to put our model into the broader context of the mathematical literature
We will informally name this phenomenon the magnetic pressure problem: Definition 1. (Magnetic pressure problem) Introducing extra terms N (v, B) that are not divergence-free into the equation for the magnetic field B in the MHD equations will lead to the appearance of an artificial, possibly unphysical “magnetic pressure” π, i.e. it will be of the form
Summary
The magnetohydrodynamics (MHD) equations describe the dynamic motion of electrically conducting fluids. They combine the equations of motion for fluids (Navier– Stokes equations) with the field equations of electromagnetic fields (Maxwell’s equations), coupled via Ohm’s law. The equations are a macroscopic model for plasmas in that they deal with averaged quantities and assume the fluid to be a continuum with frequent collisions. Both approximations are not met in hot plasmas. The MHD equations provide a good description of the lowfrequency, long-wavelength dynamics of real plasmas. Mathematics Subject Classification: 76W05, 76S05, 35K91, 76D03 Keywords: Tamed MHD equations, Magnetohydrodynamics, MHD equations, Porous media, Suitable weak solution
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