Global existence of large solutions to initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas. (English) Zbl 0579.35052

Kawohl, Bernhard

J. Differ. Equations 58, 76-103 (1985).

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)

Keywords:

large solutions; initial boundary value problem; viscous heat conducting gas; unique solution; regularity of derivatives; shocks; dissipative effect

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