# American Institute of Mathematical Sciences

January  2009, 24(1): 95-114. doi: 10.3934/dcds.2009.24.95

## Persistence of Boltzmann entropy in fluid models

 1 CMAP-CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France

Received  July 2007 Revised  December 2007 Published  January 2009

Higher order entropies are kinetic entropy estimators suggested by Enskog expansion of Boltzmann entropy. These quantities are quadratic in the density $\rho$, velocity $v$ and temperature $T$ renormalized derivatives. We investigate asymptotic expansions of higher order entropies for compressible flows in terms of the Knudsen $\epsilon_k$ and Mach $\epsilon_m$ 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 ||$\log \rho$||BMO,$\quad$ $\epsilon_m$||$v/\sqrt{T}$|| L ,$\quad$ ||$\log T$||$BMO$,$\quad$ $\epsilon_k$||$h\partial_{x} \rho$/$\rho$|| L , $\epsilon_k$$\epsilon_m$||$h\partial_{x} v$/$\sqrt{T}$ || L , $\epsilon_k$||$h\partial_{x}T$/$T$|| L , and $\epsilon_k^2$||$h^2\partial^2_x T$/$T$|| L are small enough, where $h = 1/(\rho T^{(1/2) -x)}$ is a weight associated with the dependence on density and temperature of the mean free path.
Citation: Vincent Giovangigli. Persistence of Boltzmann entropy in fluid models. Discrete & Continuous Dynamical Systems, 2009, 24 (1) : 95-114. doi: 10.3934/dcds.2009.24.95
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