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

Bernard Ducomet·Alexander Zlotnik
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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 $p(η,θ)=p_0(η)+p_1(η)θ$ linear with respect to the temperature $θ$, but we admit rather general nonmonotone functions $p_0$ and $p_1$ of $η$, which allows us to treat various physical models of nuclear fluids (for which $p$ and $η$ 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 $η$, together with its asymptotic behaviour as $t\to \infty$. Namely, we establish the stabilization pointwise and in $L^q$ for $η$, in $L^2$ for $θ$, and in $L^q$ for $v$ (the velocity), for any $q\in[2,\infty)$.

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Bernard Ducomet, Alexander Zlotnik. 2002-06-05. Stabilization for equations of one-dimensional viscous compressible heat-conducting media with nonmonotone equation of state. https://arxiv.org/abs/math-ph/0206005

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