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Asymptotic behavior of solutions for the equations of a viscous heat- conductive gas. (English) Zbl 0624.76097

We study the asymptotic behavior of solutions to the initial value problem for the equations of a viscous heat-conductive gas in Lagrangian coordinates: (*) \(v_ t-u_ x=0,\quad u_ t+p_ x=(\mu u_ x/v)_ x,\quad (e+u^ 2/2)_ t+(pu)_ x=(k\theta_ x/v+\mu uu_ x/v)_ x\), where the unknowns \(v>0\), u and \(\theta >0\) represent the specific volume, the velocity and the absolute temperature of the gas. The coefficients of viscosity and heat-conductivity, \(\mu\) and k, are assumed to be positive constants. We denote the initial function for (*) by \(U_ 0(x)=(v_ 0\), \(u_ 0\), \(\theta_ 0)(x)\) and put \(U_{\pm}=U_ 0(\pm \infty)\). We consider the case where \(U_-\) is connected to \(U_+\) by only rarefaction waves, and show that the solution of (*) converges to the weak solution of the Riemann problem for the inviscid equations.

MSC:

76N15 Gas dynamics (general theory)
80A20 Heat and mass transfer, heat flow (MSC2010)
35Q99 Partial differential equations of mathematical physics and other areas of application
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