Persistence of Boltzmann entropy in fluid models

Higher order entropies are kinetic entropy estimators suggested by Enskog expansion of Boltzmann entropy. These quantities are quadratic in the density $\rho$, velocity $v$ and temperature $T$ renormalized derivatives. We investigate asymptotic expansions of higher order entropies for compressible flows in terms of the Knudsen $\epsilon_k$ and Mach $\epsilon_m$ numbers in the natural situation where the volume viscosity, the shear viscosity, and the thermal conductivity depend on temperature, essentially in the form $T^x$. Entropic inequalities are obtained when ||$\log \rho$||BMO,$\quad$ $\epsilon_m$||$v/\sqrt{T}$|| L ^∞ ,$\quad$ ||$\log T$||$BMO$,$\quad$ $\epsilon_k$||$h\partial_{x} \rho$/$\rho$|| L ^∞ , $\epsilon_k$$\epsilon_m$||$h\partial_{x} v$/$\sqrt{T}$ || L ^∞ , $\epsilon_k$||$h\partial_{x}T$/$T$|| L ^∞ , and $\epsilon_k^2$||$h^2\partial^2_x T$/$T$|| L ^∞ are small enough, where $h = 1/(\rho T^{(1/2) -x)}$ is a weight associated with the dependence on density and temperature of the mean free path.