Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Persistence of Boltzmann entropy in fluid models

Pages: 95 - 114, Volume 24, Issue 1, May 2009      doi:10.3934/dcds.2009.24.95

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Vincent Giovangigli - CMAP-CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France (email)

Abstract: 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.

Keywords:  Entropy, Boltzmann, Kinetic, Fluid, Enskog.
Mathematics Subject Classification:  Primary: 35Q30, 76N10, 82B40.

Received: July 2007;      Revised: December 2007;      Available Online: January 2009.