Abstract
In this paper, the existence of global classical solutions is justified for the three-dimensional compressible magnetohydrodynamic (MHD) equations with vacuum. The main goal of this paper is to obtain a unique global classical solution on mathbb{R}^{3}times [0, T] with any Tin (0, infty ), provided that the initial magnetic field in the L^{3}-norm and the initial density are suitably small. Note that the first result is obtained under the condition of rho _{0}in L^{gamma }cap W^{2, q} with qin (3, 6) and gamma in (1, 6). It should be noted that the initial total energy can be arbitrarily large, the initial density allowed to vanish, and the system does not satisfy the conservation law of mass (i.e., rho _{0} notin L^{1}). Thus, the results obtained particularly extend the one due to Li–Xu–Zhang (Li et al. in SIAM J. Math. Anal. 45:1356–1387, 2013), where the global well-posedness of classical solutions with small energy was proved.
Highlights
One of the important problems in the theory of magnetohydrodynamics (MHD) is that of existence of global solutions to the equations of motion for a viscous compressible fluid
We consider the MHD system of equations for a compressible isentropic MHD flows which in the case of 3D motion has the form:
In [5], Huang et al (2012) established the global existence and uniqueness of classical solutions to the Cauchy problem for the compressible Navier–Stokes equations in 3D with smooth initial data that are of small energy
Summary
One of the important problems in the theory of magnetohydrodynamics (MHD) is that of existence of global solutions to the equations of motion for a viscous compressible fluid. In [5], Huang et al (2012) established the global existence and uniqueness of classical solutions to the Cauchy problem for the compressible Navier–Stokes equations in 3D with smooth initial data that are of small energy. In [10], Li et al (2019) were concerned with the global well-posedness and large time asymptotic behavior of strong solutions to the Cauchy problems of the Navier–Stokes equations for viscous compressible barotropic flows in 2D and 3D. For the Cauchy problem, Li et al (2013) in [9] considered the threedimensional isentropic compressible magnetohydrodynamic equations, and they proved the global well-posedness of a classical solution with small energy but possibly large oscillations, where the flow density was allowed to contain vacuum states.
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