Global weak solutions with higher regularity to the two-dimensional isentropic compressible Navier-Stokes and magnetohydrodynamic equations with far-field vacuum and unbounded density

We establish the global existence of a class of weak solutions to the isentropic compressible Navier-Stokes and magnetohydrodynamic (MHD) equations on the whole plane under a suitably small initial energy. The solutions constructed here admit far-field vacuum and unbounded densities. Moreover, they possess an intermediate regularity regime between the finite-energy weak solutions of Lions-Feireisl and the framework of Hoff. This particularly extends our previous half-plane case with Dirichlet boundary conditions (arXiv:2601.11852) to the whole-plane MHD coupling, and we also generalize the works of Hoff (Comm. Pure Appl. Math. 55 (2002), pp. 1365-1407) and Suen and Hoff (Arch. Ration. Mech. Anal. 205 (2012), pp. 27-58) by allowing vacuum states and unbounded density. Our analysis lies in a new perspective that exploits the spatial integrability of the density and the resulting integrability of the pressure, together with the specific structure of the MHD system.