  The mathematical analysis on the behavior of the entropy for viscous,
compressible, and heat conducting magnetohydrodynamic flows near the vacuum
region is a challenging problem as the governing equation for entropy is highly
degenerate and singular in the vacuum region. In particular, it is unknown
whether the entropy remains its boundedness. In the present paper, we
investigate the Cauchy problem to the three-dimensional (3D) compressible
heat-conducting magnetohydrodynamic equations with vacuum at infinity only. We
show that the uniform boundedness of the entropy and the $L^2$ regularities of
the velocity and temperature can be propagated provided that the initial
density decays suitably slow at infinity. The main tools are based on
singularly weighted energy estimates and De Giorgi type iteration techniques
developed by Li and Xin (arXiv:2111.14057) for the 3D full compressible
Navier-Stokes system. Some new mathematical techniques and useful estimates are
developed to deduce the lower and upper bounds on the entropy.
