The tamed MHD equations
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Journal of Evolution Equations
The tamed MHD equations Andre Schenke
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 R3 , 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.
1. Introduction 1.1. Magnetohydrodynamics 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. In plasma physics, 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. Nonetheless, the MHD equations provide a good description of the lowfrequency, long-wavelength dynamics of real plasmas. In this thesis, we consider the incompressible, viscous, resistive equations with homogeneous mass density and regularised variants of it. In dimensionless formulation, the MHD equations are of the following form: 1 ∂v = v − (v · ∇) v + S (B · ∇) B + ∇ ∂t Re ∂B 1 = B − (v · ∇) B + (B · ∇)v ∂t Rm div v = 0, div B = 0.
S|B|2 p+ 2
, (1)
Mathematics Subject Classification: 76W05, 76S05, 35K91, 76D03 Keywords: Tamed MHD equations, Magnetohydrodynamics, MHD equations, Porous media, Suitable weak solution.
A. Schenke
J. Evol. Equ.
Here, v = v(x, t), B = B(x, t) denote the velocity and magnetic fields, p = p(x, t) is the pressure, Re > 0, Rm > 0 are the Reynolds number and the magnetic Reynolds number and S > 0 denotes the Lundquist number (all of which are dimensionless constants). The two last equations concerning the divergence-freeness of the velocity and magnetic field are the incompressibility of the flow and Maxwell’s second equation. For simplicity, in the remainder of the paper, we set S = Rm = Re = 1. Mathematical treatment of the deterministic MHD equations reaches back to the works of Duvaut and Lions [12] and Sermange and Temam [41]. Since then, a large amount of papers have been devoted to the subject. We only mention several interesting regularity criteria [8,21,22,27] and the more recent work on non-resistive MHD equations (Rm = ∞) by C.L. Fefferman, D.S. McCormick J.C. Robinson and J.L. Rodrigo on local existence via higher-order commutator estimates [15,16]. In this paper, we want to study a regularised version of the MHD equations on the whole space R3 , which we call the tamed MHD equations (TMHD), following Röckner and Zhang [39]. They arise from (1) by adding two extra terms (the taming terms) that act as restoring forces: ∂v S|B|
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