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On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics. (English) Zbl 1295.35122
Summary: In this paper, we establish an optimal blow-up criterion for classical solutions to the incompressible resistive Hall-magnetohydrodynamic equations. We also prove two global-in-time existence results of the classical solutions for small initial data, the smallness conditions of which are given by the suitable Sobolev and the Besov norms respectively. Although the Sobolev space version is already an improvement of the corresponding result in [D. Chae et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31, No. 3, 555–565 (2014; Zbl 1297.35064)], the optimality in terms of the scaling property is achieved via the Besov space estimate. The special property of the energy estimate in terms of \(\dot B_{2,1}^s\) norm is essential for this result. Contrary to the usual MHD the global well-posedness in the \(2\frac{1}{2}\) dimensional Hall-MHD is wide open.

35B44 Blow-up in context of PDEs
35L60 First-order nonlinear hyperbolic equations
35K55 Nonlinear parabolic equations
76W05 Magnetohydrodynamics and electrohydrodynamics
35Q35 PDEs in connection with fluid mechanics
35Q60 PDEs in connection with optics and electromagnetic theory
Full Text: DOI arXiv
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