Local well-posedness in critical spaces for the compressible MHD equations

Abstract

In this paper, we prove local well-posedness in critical Besov spaces for the compressible MHD equations in under the assumptions that the initial density is bounded above and bounded away from zero. The proof relies on uniform estimates for a mixed hyperbolic/parabolic linear system with a convection term.

Keywords:

• compressible MHD equations
• Besov spaces
• critical spaces
• Littlewood-Paley theory
• local well-posedness

AMS Subject Classifications:

• 76W05
• 76N10

Acknowledgments

This work was done when D. Bian pursued her master degree at Henan Polytechnic University.

Notes

1 The research of D. Bian was partially supported by National Natural Science Foundation of China [grant number 11071057] and [grant number 11271052] and Tian Yuan Special Foundation [grant number 11226029]. The research of B. Yuan was partially supported by the National Natural Science Foundation of China [grant number 11071057]; Innovation Scientists and Technicians Troop Construction Projects of Henan Province [grant number 104100510015].