Singularity formation to the two-dimensional non-resistive compressible magnetohydrodynamic equations in a bounded domain

The question of whether the two-dimensional compressible magnetohydrodynamic equations without magnetic diffusion can develop a finite-time singularity from smooth initial data is a challenging problem in fluid dynamics and mathematics. In this paper, we derive a regularity criterion which shows that the strong solution exists globally if the density and the magnetic field are bounded from above. Our method relies on critical Sobolev inequalities of logarithmic type.