Stabilization for equations of one-dimensional viscous compressible heat-conducting media with nonmonotone equation of state

Abstract

We consider the Navier-Stokes system describing motions of viscous compressible heat-conducting and "self-gravitating" media. We use the state function of the form $p(\eta,\theta)=p_0(\eta)+p_1(\eta)\theta$ linear with respect to the temperature $\theta$, but we admit rather general nonmonotone functions $p_0$ and $p_1$ of $\eta$, which allows us to treat various physical models of nuclear fluids (for which $p$ and $\eta$ are the pressure and specific volume) or thermoviscoelastic solids. For an associated initial-boundary value problem with "fixed-free" boundary conditions and possibly large data, we prove a collection of estimates independent of time interval for solutions, including two-sided bounds for $\eta$, together with its asymptotic behaviour as $t\to \infty$. Namely, we establish the stabilization pointwise and in $L^q$ for $\eta$, in $L^2$ for $\theta$, and in $L^q$ for $v$ (the velocity), for any $q\in[2,\infty)$.