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

TL;DR

This paper proves the long-term stabilization and boundedness of solutions for a viscous, heat-conducting compressible fluid model with nonmonotone pressure functions, applicable to various physical media including nuclear fluids and thermoviscoelastic solids.

Contribution

It establishes uniform estimates and asymptotic behavior for solutions of a complex Navier-Stokes system with nonmonotone pressure functions, extending the understanding of stability in such media.

Findings

Solutions are bounded uniformly in time.

Pressure, specific volume, temperature, and velocity stabilize asymptotically.

Results apply to large initial data and general physical models.

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 $q\in[2,\infty)$.