Convergence of the viscosity method for a nonstrictly hyperbolic conservation law
The author considers the existence of global weak solutions for nonlinear hyperbolic conservation laws \[ v_ t+(vu)_ x=0,\quad u_ t+\left( {1\over 2} u^ 2+ \int^ v s(s+\delta)^{r-3}ds \right)_ x=0 \tag{1} \] with initial data (2) \((v(x,0)\), \(u(x,0))=(v_ 0(x)\), \(u_ 0(x))\), where \(\delta,r\) are positive constants and \(r>3\). When \(\delta=0\), (1) is motivated by the isentropic equation of gas dynamics for a polytropic gas. The global weak solutions of which had been solved for the case of \(1<r<3\) by using the Glimm difference scheme by DiPerna. In the present paper system (1) is studied with bounded measurable initial data (2) by using the established technique of compensated compactness. Through an analysis of progressing entropy waves, we establish a convergence theorem for the method of artificial viscosity applied to the system (1) and obtain the existence of the global weak solutions for the Cauchy problem (1), (2).
isentropic equation of gas dynamics for a polytropic gas
scientific article; zbMATH DE number 165645