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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 \(Ma=\epsilon\), \(Fr=\sqrt{\epsilon}\), \(Al=\sqrt{\epsilon}\) where \(\epsilon\) 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 \(\epsilon \to 0\) is studied in great detail in both bounded and unbounded domains.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35B45 A priori estimates in context of PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
35Q35 PDEs in connection with fluid mechanics
35B25 Singular perturbations in context of PDEs
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