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Due to the highly degeneracy and singularities of the entropy equation, thephysical entropy for viscous and heat conductive polytropic gases behavesingularly in the presence of vacuum and it is thus a challenge to study itsdynamics. It is shown in this paper that the uniform boundedness of the entropyand the inhomogeneous Sobolev regularities of the velocity and temperature canbe propagated for viscous and heat conductive gases in $\mathbb R^3$, providedthat the initial vacuum occurs only at far fields with suitably slow decay ofthe initial density. Precisely, it is proved that for any strong solution tothe Cauchy problem of the heat conductive compressible Navier--Stokesequations, the corresponding entropy keeps uniformly bounded and the $L^2$regularities of the velocity and temperature can be propagated, up to theexisting time of the solution, as long as the initial density vanishes only atfar fields with a rate no more than $O(\frac{1}{|x|^2})$. The main tools aresome singularly weighted energy estimates and an elaborate De Giorgi typeiteration technique. The De Giorgi type iterations are carried out to differentequations in establishing the lower and upper bounds of the entropy.

Propagation of uniform boundedness of entropy and inhomogeneous  regularities for viscous and heat conductive gases with far field vacuum in  three dimensions