Global existence of smooth solutions and convergence to Barenblatt solutions for the physical vacuum free boundary problem of compressible Euler equations with damping (Q2812292)

The authors consider the free boundary problem for the one-dimensional Euler equations with damping for compressible isentropic flows where the system NEWLINE\[NEWLINE\rho_t+(\rho u)_x=0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE(\rho u)_t+(\rho^\gamma+\rho u^2)_x=-\rho u,NEWLINE\]NEWLINE is satisfied for \(x\in(x_-(t),x_+(t))\), and \(\rho=0\) on \(\Gamma(t)=\{(x,t): \;x=x_{\pm}(t),\;t>0\}\), \(\dot\Gamma(t)=u(\gamma(t),t)\). The main result is the global in time existence of solutions and asymptotic equivalence of smooth solutions to the Barenblatt self-similar solutions of the equation \(\rho_t=(\rho^\gamma)_{xx}\) with their parameters determined by the initial total mass \(\int_{x_-(0)}^{x_+(0)} \rho_0(x)dx\). This is an extension of results by \textit{T.-P. Liu} [Japan J. Ind. Appl. Math. 13, No. 1, 25--32 (1996; Zbl 0865.35107)] for particular solutions of that system, and shows the asymptotic stability of Barenblatt profiles. Here, the sound speed is \(\frac12\)-Hölder continuous across vacuum boundaries which makes the problem inaccessible for standard methods of symmetric hyperbolic systems.