The method of compensated compactness applied to the analysis of one- dimensional gasdynamics equations (Q1822272)

The author studies the system \[ (1)\quad \frac{\partial u}{\partial t}+\frac{\partial}{\partial x}(\frac{1}{2} u^ 2\frac{\gamma}{\gamma -1} \rho^{\gamma -1})=\epsilon \frac{\partial^ 2u}{\partial x^ 2},\quad \frac{\partial \rho}{\partial t}+\frac{\partial}{\partial x} (u\rho)=\epsilon \frac{\partial^ 2\rho}{\partial x^ 2} \] with periodic and continuous initial data \[ (2)\quad u_{| t=0}=u_ 0(x),\quad \rho_{| t=0}=\rho_ 0(x)>0. \] When \(\epsilon =0\), this system describes the one-dimensional gas dynamics. Provided the solutions \((u^{\epsilon},\rho^{\epsilon})\), \(\epsilon >0\), of (1), (2) satisfy some a priori estimates the author proves that there exists a subsequence \((u^{\epsilon (n)},\rho^{\epsilon (n)})\), \(\epsilon\) (n)\(\to 0\), which tends weakly to a generalized solution of the problem (1), (2) with \(\epsilon =0\).