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Norm inflation for incompressible magneto-hydrodynamic system in \(\dot B^{-1,\infty}_\infty\). (English) Zbl 1357.76099

Summary: Based on the construction of J. Bourgain and N. Pavlović [J. Funct. Anal. 255, No. 9, 2233–2247 (2008; Zbl 1161.35037)], we show that the solutions to the Cauchy problem for the three-dimensional incompressible magneto-hydrodynamics system can develop different types of norm inflations in \(\dot B_{\infty}^{-1,\infty}\). In particular the magnetic field can develop norm inflation in a short time even when the velocity remains small and vice versa. Efforts are made to present a very expository development of the ingenious construction of Bourgain and Pavlović [loc. cit.].

MSC:

76W05 Magnetohydrodynamics and electrohydrodynamics
35Q35 PDEs in connection with fluid mechanics
35Q60 PDEs in connection with optics and electromagnetic theory

Citations:

Zbl 1161.35037
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