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On the incompressible limits for the full magnetohydrodynamics flows. (English) Zbl 1242.35051
This paper studies the incompressible limits for weak solutions for the full magentohydrodynamics flows in bounded and unbounded domains. In the model, various physically acceptable assumptions are made, e.g., the viscous stress tension is determined through Newton’s rheological law, the heat flux is given by Fourier’s law etc., and a scaling of the dimensionless parameters of the Mach, Froude and Alfven number is assumed according to which $Ma=ϵ$, $Fr=\sqrt{ϵ}$, $Al=\sqrt{ϵ}$ where $ϵ$ is small. A variational formulation is provided for the full problem and specific conditions for the data of the problem so that this formulation holds are stated. The limit as $ϵ\to 0$ is studied in great detail in both bounded and unbounded domains.
##### MSC:
 35B40 Asymptotic behavior of solutions of PDE 76N10 Compressible fluids, general 35B45 A priori estimates for solutions of PDE 76W05 Magnetohydrodynamics and electrohydrodynamics 35Q35 PDEs in connection with fluid mechanics 35B25 Singular perturbations (PDE)