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Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. (English) Zbl 1465.35331

Summary: We prove some results on the stability of slow stationary solutions of the MHD equations in two- and three-dimensional bounded domains for external force fields that are asymptotically autonomous. Our results show that weak solutions are asymptotically stable in time in the \(L^2\)-norm. Further, assuming certain regularity hypotheses on the problem data, strong solutions are asymptotically stable in the \(H^1\) and \(H^2\)-norms.

MSC:

35Q30 Navier-Stokes equations
76E25 Stability and instability of magnetohydrodynamic and electrohydrodynamic flows
35Q60 PDEs in connection with optics and electromagnetic theory
76W05 Magnetohydrodynamics and electrohydrodynamics
35B40 Asymptotic behavior of solutions to PDEs
35D30 Weak solutions to PDEs
35D35 Strong solutions to PDEs
35B35 Stability in context of PDEs
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