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 ρ, velocity v and temperature T renormalized derivatives. We investigate asymptotic expansions of higher order entropies for compressible flows in terms of the Knudsen ǫk and Mach ǫm numbers in the natural situation where the volume viscosity, the shear viscosity, and the thermal conductivity depend on temperature, essentially in the form T . Entropic inequalities are obtained when ‖ log ρ‖BMO , ǫm‖v/ √ T‖L∞ , ‖ log T‖BMO , ǫk‖h∂xρ/ρ‖L∞ , ǫkǫm‖h∂xv/ √ T‖L∞ , ǫk‖h∂xT/T‖L∞ , and ǫk‖h∂ xT/T‖L∞ are small enough, where h = 1/ρT 1 2 −κ is a weight associated with the dependence on density and temperature of the mean free path. AMS subject classifications : 35K, 76N, 82A40.