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Global existence of large solutions to initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas. (English) Zbl 0579.35052
The initial boundary value problem considered here is \[ u_ t-v_ x=0,\quad v_ t-\sigma_ x=0,\quad (e+(1/2)v^ 2)t-(\sigma v-q)_ x=0,\quad \eta_ t+(q/\theta)_ x\geq 0 \] with \(u(x,0)=u_ 0(x)\), \(v(x,0)=v_ 0(x)\), \(\theta (x,0)=\theta_ 0(x)\) on [0,1]. The boundary conditions are \(q(0,t)=q(1,t)=0\) if \(t\geq 0\) and either \(\sigma (0,t)=\sigma (1,t)=0\) or \(v(0,t)=v(1,t)=0\). This is a model of a viscous heat conducting gas. Under certain conditions it is proved that there is a unique solution \(\{\) u(x,t),v(x,t),\(\theta\) (x,t)\(\}\) on [0,1]\(\times [0,\infty)\) with \(u,v>0\) and satisfying regularity of derivatives for each finite time. In other words, shocks do not develop because the dissipative effect of the nonlinearities is sufficient to prevent them.
Reviewer: E.Barron

MSC:
35L60 First-order nonlinear hyperbolic equations
35L67 Shocks and singularities for hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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References:
[1] Becker, E, Gasdynamik, (1966), Teubner Verlag Stuttgart · Zbl 0139.20901
[2] Dafermos, C.M; Hsiao, L, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity, Nonlinear anal. theory methods appl., 6, 435-454, (1982) · Zbl 0498.35015
[3] Dafermos, C.M, Global smooth solutions to the initial boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity, SIAM J. math. anal., 13, 397-408, (1982) · Zbl 0489.73124
[4] Friedman, A, Partial differential equations of parabolic type, (1964), Prentice-Hall Englewood Cliffs, N.J · Zbl 0144.34903
[5] Friedman, A, Partial differential equations, (1976), Krieger New York
[6] Kazhykhov, A.V, Sur la solubilité globale des problèmes monodimensionnels aux valeurs initiales-limitées pour LES équations d’un gaz visqueux et calorifère, C. R. acad. sci., Paris ser. A, 284, 317-320, (1977) · Zbl 0355.35071
[7] Kazhikhov, A.V; Shelukhin, V.V; Kazhikhov, A.V; Shelukhin, V.V, Unique global solution with respect to time of initial boundary value problem for one-dimensional equations of a viscous gas, Prikl. mat. mekh., J. appl. math. mech., 41, 273-282, (1977), English translation
[8] Ladyzenskaya, O.A; Solonnikov, V.A; Ural’ceva, N.N, Linear and quasilinear equations of parabolic type, (1968), Amer. Math. Soc Providence, R.I
[9] Matsumura, A; Nishida, T, Initial boundary value problems for the equations of motion of compressible viscous fluids, () · Zbl 0543.76099
[10] Partington, J.R, An advanced treatise on physical chemistry, (1949), Longmans Green, London
[11] Protter, M.H; Weinberger, H.F, Maximum principles in differential equations, (1967), Prentice-Hall Englewood Cliffs, N.J · Zbl 0153.13602
[12] Solonnikov, V.A; Kazhikhov, A.V, Existence theorems for the equations of motion of a compressible viscous fluid, Ann. rev. fluid mech., 13, 79-95, (1981) · Zbl 0492.76074
[13] Zel’dovich, Y.B; Raizer, Y.P, ()
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