Global solvability of large initial data to three-dimensional compressible MHD equations with density-dependent viscosities

This paper studies the three-dimensional isentropic compressible magnetohydrodynamic equations with density-dependent viscosities, considering both the Cauchy problem and periodic problem. We prove that the strong solution exists globally provided that the initial density is large enough. This is a result that generalizes previous classical results which the solutions have small perturbations from the resting state. This result can also be seen as an extension of the results for Yu [Math. Methods Appl. Sci., 46 (2023) 10123–10136] and Huang–Li–Zhang [arXiv:2408.04305, 2024], who established similar results for the compressible Navier–Stokes equations. At the same time, this article improves on Li–Lu–Shang’s recent result [arXiv:2408.04995, 2024] on the compressible MHD equations. The key idea of the proof is to establish an effective energy functional about density, velocity and magnetic field, which combines Huang–Li–Zhang’s article [arXiv:2408.04305, 2024] on the barotropic compressible Navier–Stokes equations with Fan–Gu–Huang’s techniques [Discrete Contin. Dyn. Syst., 45 (2025) 2628–2649] on the non-homogeneous incompressible MHD equations.