Higher order entropies for compressible fluid models

We investigate higher order entropies for compressible fluid models and related a priori estimates. 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 governing equations of higher order entropy correctors and related differential inequalities in the natural situation where the volume viscosity, the shear viscosity, and the thermal conductivity depend on temperature, essentially in the form T^N, as given by the kinetic theory of gases. Entropic inequalities are established when ||log ρ|| _BMO, $\Vert v/√{T}\Vert-{L^{\infty}}$, ||log T||_BMO, ||h∂_x ρ/ρ ||_L ^∞, $Vert h∂-x v/√{T} \Vert-{L{∞} $, || h∂_xT/T||_L^∞, and $\Vert h∂2-x T/T \Vert-{L^{\infty}}$ are small enough, where $h = 1/(ρ T{1}{2} -\varkappa})$ is a weight associated with the dependence of the local mean free path on density and temperature. As an example of application, we investigate global existence of solutions when the initial values log(ρ₀/ρ_∞), $v-0/√{T-0}$, and log(T₀/T_∞) are small enough in appropriate spaces.