Global well-posedness to three-dimensional full compressible magnetohydrodynamic equations with vacuum

Abstract

This paper studies the Cauchy problem for three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic equations with vacuum as far field density. We prove the global existence and uniqueness of strong solutions provided that the quantity $\Vert {\rho }_0{\Vert }_{L^{\infty }}+\Vert b_0{\Vert }_{L^3}$ is suitably small and the viscosity coefficients satisfy $3\mu >\lambda$. Here, the initial velocity and initial temperature could be large. The assumption on the initial density do not exclude that the initial density may vanish in a subset of ${\mathbb{R}}^3$ and that it can be of a nontrivially compact support. Our result is an extension of the works of Fan and Yu \cite{FY09} and Li et al. \cite{LXZ13}, where the local strong solutions in three dimensions and the global strong solutions for isentropic case were obtained, respectively. The analysis is based on some new mathematical techniques and some new useful energy estimates. This paper can be viewed as the first result concerning the global existence of strong solutions with vacuum at infinity in some classes of large data in higher dimension.

arXiv comment23 pages

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Cited by in corpus (3)

• Local well-posedness to the 2D Cauchy problem of full compressible magnetohydrodynamic equations with vacuum at infinity
• Entropy-bounded solutions to the 3D compressible heat-conducting magnetohydrodynamic equations with vacuum at infinity
• Global Strong and Weak Solutions to the Initial-boundary-value Problem of 2D Compressible MHD System with Large Initial Data and Vacuum