A blow-up criterion for three-dimensional compressible magnetohydrodynamic equations with variable viscosity

Xin Zhong

doi:10.3934/dcdsb.2018318

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2019, Volume 24, Issue 7: 3249-3264. Doi: 10.3934/dcdsb.2018318

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A blow-up criterion for three-dimensional compressible magnetohydrodynamic equations with variable viscosity

Xin Zhong

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received: October 2017

Early access: January 2019

Published: May 2019

Supported by Chongqing Research Program of Basic Research and Frontier Technology (No. cstc2018jcyjAX0049), the Postdoctoral Science Foundation of Chongqing (No. xm2017015), and China Postdoctoral Science Foundation (Nos. 2018T110936, 2017M610579).

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Abstract

We are concerned with the breakdown of strong solutions to the three-dimensional compressible magnetohydrodynamic equations with density-dependent viscosity. It is shown that for the initial density away from vacuum, the strong solution exists globally if the gradient of the velocity satisfies $ \|\nabla{\bf{u}}\|_{L^{2}(0,T;L^\infty)}<\infty $. Our method relies upon the delicate energy estimates and elliptic estimates.

Keywords:

Compressible magnetohydrodynamic equations,
blow-up,
variable viscosity.

Mathematics Subject Classification: 76W05, 35B65.

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References

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