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 p, velocity v and temperature T renormalized derivatives. We investigate asymptotic expansions of higher order entropies for compressible flows in terms of the Knudsen ε_K and Mach ε_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 ρ|| BMO, ε_M||v/√T||L∞, ||logT||BMO. ε _K||h∂_x ρ/ρ/||L∞, ε _Kε_M||h∂_x v/√T||L∞, ε_K||h∂_xT/T||L∞, and ε ²_K||h² ∂²_xT/T || L∞ are small enough, where h = l/(ρT^1/2-x) is a weight associated with the dependence on density and temperature of the mean free path.