On Local Strong Solutions to the Cauchy Problem of Two-Dimensional Density-Dependent Magnetohydrodynamic Equations with Vacuum

Abstract

This paper concerns the Cauchy problem of the nonhomogeneous incompressible magnetohydrodynamic (MHD) equations on the whole two-dimensional (2D) space with vacuum as far field density. In particular, the initial density can have compact support. We prove that the 2D Cauchy problem of the nonhomogeneous incompressible MHD equations admits a unique local strong solution provided the initial density and the initial magnetic decay not too slow at infinity.

arXiv comment22 pages. arXiv admin note: text overlap with arXiv:1306.4752 by other authors

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• Global Existence and Large Time Asymptotic Behavior of Strong Solutions to the Cauchy Problem of 2D Density-Dependent Navier-Stokes Equations with Vacuum
• On classical solutions to the Cauchy problem of the 2D compressible non-resistive MHD equations with vacuum
• Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows with vacuum
• Global Existence and Large Time Asymptotic Behavior of Strong Solutions to the Cauchy Problem of 2D Density-Dependent Magnetohydrodynamic Equations with Vacuum
• Global Strong Solutions to the Compressible Magnetohydrodynamic Equations with Slip Boundary Conditions in 3D Bounded Domains
• Local well-posedness to the 2D Cauchy problem of full compressible magnetohydrodynamic equations with vacuum at infinity
• A Remark on stress of a spatially uniform dislocation density field
• Supercongruences for Almkvist--Zudilin sequences
• On formation of singularity of the full compressible magnetohydrodynamic equations with zero heat conduction
• Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum
• A blow-up criterion for strong solutions to three-dimensional compressible magnetohydrodynamic equations
• Existence theorems for the Cauchy problem of 2D nonhomogeneous incompressible non-resistive MHD equations with vacuum
• Global Strong and Weak Solutions to the Initial-boundary-value Problem of 2D Compressible MHD System with Large Initial Data and Vacuum
• Strong solutions to the Cauchy problem of the two-dimensional non-baratropic non-resistive magnetohydrodynamic equations with zero heat conduction
• Global small solutions to a special $2\frac{1}{2}$-D compressible viscous non-resistive MHD system