Higher order entropies

摘要

Higher order entropies are kinetic entropy estimators for fluid models. These quantities are quadratic in the velocity and temperature derivatives and have temperature dependent coefficients. We investigate governing equations for higher order entropies and related a priori estimates in the natural situation where viscosity and thermal conductivity depend on temperature. We establish positivity of higher order derivative source terms in these governing equations provided that parallel to log T parallel to(BMO) + parallel to upsilon / root T parallel to L-infinity is small enough. The temperature factors renormalizing temperature and velocity derivatives then yield majorization of lower order convective terms only when the temperature dependence of transport coefficients is taken into account according to the kinetic theory. In this situation, we obtain entropic principles for higher order entropies of arbitrary order. As an application, we investigate a priori estimates and global existence of solutions when the initial values log(T-0/T-infinity) and upsilon / root T-0 are small enough in appropriate spaces.