Global Well-Posedness and Asymptotic Behavior of Strong Solutions to an Initial-Boundary Value Problem of 3D Full Compressible MHD Equations

This paper is concerned with an initial-boundary value problem of full compressible magnetohydrodynamics (MHD) equations on 3D bounded domains subject to non-slip boundary condition for velocity, perfectly conducting boundary condition for magnetic field, and homogeneous Dirichlet boundary condition for temperature. The global well-posedness of strong solutions with initial vacuum is established and the exponential decay estimates of the solutions are obtained, provided the initial total energy is suitably small. More interestingly, it is shown that for \(p\in (3,6)\) $p\in (3,6)$ , the \(L^p\) $L^p$ -norm of the gradient of density remains uniformly bounded for all \(t\ge 0\) $t\ge 0$ . This is in sharp contrast to that in (Chen et al. in Global well-posedness of full compressible magnetohydrodynamic system in 3D bounded domains with large oscillations and vacuum. arXiv:2208.04480, Li et al. in Global existence of classical solutions to full compressible Navier–Stokes equations with large oscillations and vacuum in 3D bounded domains. arXiv:2207.00441), where the exponential growth of the gradient of density in \(L^p\) $L^p$ -norm was explored.