Asymptotics of Higher Order Entropies

Higher order entropies are kinetic entropy estimators for fluid models. These quantities are quadratic in the velocity v and temperature T derivatives and have temperature dependent coefficients. We investigate asymptotic expansions of higher order entropies for incompressible flows in terms of the Knudsen ǫk and Mach ǫm numbers. The correspoding entropic inequalities are obtained when ‖ log T‖BMO + ǫm‖v/ √ T‖L∞ is small enough, provided that the temperature dependence of the thermal conductivity λ and the viscosity η is that given by the kinetic theory. As an example of application of higher order entropic estimates we establish an existence theorem for small Mach number flows.