Stability properties of regular flows of heat-conducting compressible fluids

Abstract

Assume that a viscous heat-conducting fluid is moving due to the action of small external forces. We prove the following results :

i) In any domain $\Omega$, continuous dependence of the rest state $S$;

ii) In unbounded domains $\Omega$: a) uniqueness of the state of the vacuum ; b) summability, in the time interval $(0, \infty )$, for the rate of deformation tensor, for the kineticenergy and for the $L^{2}$-norm of the pressure $p$ for barotropic processes $p = k\rho ^{\gamma}$, $\gamma \geq 1$, when the initial densitity $r(x)$ is supposed summable in $L^{1}(Q)$; c) summability in thetime interval $(0, \infty )$ for the rate of deformation tensor and for the $L^{2}$-norm of the perturbance $\sigma$ to the density $\rho _{0}$, of a barotropic flow, when $r(x)$ is supposed to have apositive infimum;

iii) In bounded domains $\Omega$, exponential decay of the $L^{2}$-norm of the pertubance to $S$ for arbitrary large initial data.