Global Weak Solutions for the Cauchy Problem to One-Dimensional Heat-Conductive MHD Equations of Viscous Non-resistive Gas
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Magnetohydrodynamics is concerned with the mutual interactions between moving, electrically conductive fluids and the magnetic field. It is widely applied in astrophysics, thermonuclear reactions and industry. The Cauchy problem to one-dimensional heat-conductive magnetohydrodynamic equations of viscous non-resistive gas is considered under the framework of Lagrangian coordinates. Based on the crucial upper and lower bounds of the density, we first obtain global well-posedness of strong solutions with regular initial data. Then existence of global weak solutions is established via the compactness analysis and the method of weak convergence. Stability of weak solutions is also verified by making full use of the specific mathematical structure of the equations. All results are obtained without any restriction to the size of the initial data.
KeywordsNon-resistive MHD equations Global weak solutions Cauchy problem
Mathematics Subject Classification35M10 35D30 35K55
The research of Y. Li is supported by NSF of China under grant number 11571167 and Postgraduate Research and Practice Innovation Program of Jiangsu Province under grant number KYCX 18-0028. The authors thank Professor Y. Sun for helpful discussions. In addition, the authors are indebted to the anonymous reviewers for providing valuable comments on the manuscript.
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