Long time existence and incompressible limit for non-isentropic slightly compressible magnetohydrodynamic equations in bounded domains

We study the long time existence of classical solutions to the non-isentropic, slightly compressible magnetohydrodynamic equations in three-dimensional bounded domains. The velocity and temperature satisfy Dirichlet boundary conditions, while the magnetic field obeys the perfectly conducting boundary condition. Assuming the smooth solution of the incompressible system with large initial data exists on an arbitrary time interval [0, T], we prove that a solution of the slightly compressible system given by a small perturbation thereof exists on [0, T] and converges uniformly to that of the incompressible system. The size and lifespan of the solution to the compressible system depend only on those of the incompressible system. In contrast to the works of Fan et al. [J. Differ. Equations 410, 578–618 (2024)] and Ju et al. [Stud. Appl. Math. 154, e70066 (2025)], our analysis employs an interior-boundary energy estimate framework to overcome the difficulties from Dirichlet boundary conditions, which is the novelty of this paper. Our method also applies to other singular limit problems in bounded domains.