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Generalized MHD equations. (English) Zbl 1057.35040
The paper deals with the \(d\)-dimensional generalized MHD (GMHD) equations \[ \begin{aligned} &\partial_{t}u + u\cdot \nabla u =-\nabla P + b \cdot \nabla b - \nu (-\Delta)^{\alpha} u,\\ &\partial_{t}b + u \cdot \nabla b= b \cdot \nabla u - \eta (-\Delta)^{\beta} b, \end{aligned} \] where the fractional power \((-\Delta)^{\alpha}\) is defined via a Fourier transform and where the spatial domain \(\Omega\) is either the whole space \(\mathbb R^{d}\) or the Torus \(\mathbb{T}^{d}\). Depending on the parameters \(\nu\), \(\eta\), \(\alpha\) and \(\beta\) the author investigates three major cases: (1) \(\nu >0\) and \(\eta >0\); (2) \(\nu=0\) and \(\eta >0\); (3) \(\nu = \eta = 0\). Depending on these cases, several existence, uniqueness and in particular regularity results are established.

35Q35 PDEs in connection with fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
Full Text: DOI
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