On existence of global classical solutions to the 3D compressible MHD equations with vacuum

Abstract

AbstractIn 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 $\mathbb{R}^{3}\times [0, T]$ R 3 × [ 0 , T ] with any $T\in (0, \infty )$ T ∈ ( 0 , ∞ ) , provided that the initial magnetic field in the $L^{3}$ L 3 -norm and the initial density are suitably small. Note that the first result is obtained under the condition of $\rho _{0}\in L^{\gamma }\cap W^{2, q}$ ρ 0 ∈ L γ ∩ W 2 , q with $q\in (3, 6)$ q ∈ ( 3 , 6 ) and $\gamma \in (1, 6)$ γ ∈ ( 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., $\rho _{0} \notin L^{1}$ ρ 0 ∉ 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.