On the outer pressure problem of the one-dimensional polytropic ideal gas. (English) Zbl 0665.76076

The author considers the one-dimensional motion of a polytropic ideal gas on the interval [0,1] with adiabatic ends under pressure P(t). In earlier work, he showed that if \(P(t)\equiv 0\) then the specific volume u(x,t) diverges more rapidly than a logarithmic power, as \(t\to \infty\). In the present paper, it is assumed that \[ \inf_{t\in [0,+\infty)}P(t)>0,\quad TV(P)\equiv \int^{\infty}_{0}| P'(t)| dt<\infty. \] It is then proved that \(\lim_{t\to \infty}\int^{t}_{0}P'(\tau)\int^{1}_{0}u(x,\tau)dxd\tau\) exists, and that as \(t\to \infty\) the solution \((u,v,\theta)\) converges in \(W^{1,2}(0,1)\cap C[0,1]\) to a stationary state \((\bar u,0,{\bar \theta}\)), with \(\bar u,\) \({\bar \theta}\) explicitly known. The result improves on an earlier one of that type due to A. V. Kazhikhov [Boundary value problems for equations of hydrodynamics, Novosibirsk, 50 (1981); S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary value problems in the mechanics of inhomogeneous fluids (1983; Zbl 0568.76001)].
Reviewer: R.Finn


76N15 Gas dynamics (general theory)
35L65 Hyperbolic conservation laws
35Q99 Partial differential equations of mathematical physics and other areas of application
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