Global variational solutions to the compressible magnetohydrodynamic equations

We prove the existence of globally defined variational solutions to the compressible magnetohydrodynamic (MHD) equations with the coefficients depending on the temperature. As a by-product, we give a simple proof for the nonexistence of nontrivial weak time-periodic solutions by the entropy principle of Clausius–Duhem and a new Poincaré-type inequality.