Higher-Order Global Regularity of an Inviscid Voigt-Regularization of the Three-Dimensional Inviscid Resistive Magnetohydrodynamic Equations

摘要

We prove existence, uniqueness, and higher-order global regularity of strongsolutions to a particular Voigt-regularization of the three-dimensionalinviscid resistive Magnetohydrodynamic (MHD) equations. Specifically, thecoupling of a resistive magnetic field to the Euler-Voigt model is introducedto form an inviscid regularization of the inviscid resistive MHD system. Theresults hold in both the whole space $R^3$ and in the context of periodicboundary conditions. Weak solutions for this regularized model are alsoconsidered, and proven to exist globally in time, but the question ofuniqueness for weak solutions is still open. Since the main purpose of thisline of research is to introduce a reliable and stable inviscid numericalregularization of the underlying model we, in particular, show that thesolutions of the Voigt regularized system converge, as the regularizationparameter $lphamaps0$, to strong solutions of the original inviscidresistive MHD, on the corresponding time interval of existence of the latter.Moreover, we also establish a new criterion for blow-up of solutions to theoriginal MHD system inspired by this Voigt regularization. This type ofregularization, and the corresponding results, are valid for, and can also beapplied to, a wide class of hydrodynamic models.