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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.

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