Self-similar generalized Riemann problems for the 1-D isothermal Euler system

Abstract

We consider self-similar solutions to the 1-dimensional isothermal Euler system for compressible gas dynamics. For each\(\beta \in {\mathbb {R}}\), the system admits solutions of the form $$\begin{aligned} \rho (t,x)=t^\beta \Omega (\xi )\qquad u(t,x)=U(\xi )\qquad \qquad \textstyle \xi =\frac{x}{t}, \end{aligned}$$where \(\rho \) and u denote the density and velocity fields. The ODEs for \(\Omega \) and U can be solved implicitly and yield the solution to generalized Riemann problems with initial data $$\begin{aligned} \rho (0,x)=\left\{ \begin{array}{ll} R_l |x|^\beta &{} x<0\\ R_rx^\beta &{} x>0 \end{array}\right. \qquad u(0,x)=\left\{ \begin{array}{ll} U_l &{} x<0\\ U_r &{} x>0, \end{array}\right. \end{aligned}$$where \(R_l,\, R_r>0\) and \(U_l,\ U_r\) are arbitrary constants. For \(\beta \in (-1,0)\), the data are locally integrable but unbounded at \(x=0\), while for \(\beta \in (0,1)\), the data are locally bounded and continuous but with unbounded gradients at \(x=0\). Any (finite) degree of smoothness of the data is possible by choosing \(\beta >1\) sufficiently large and \(U_l=U_r\). (The case \(\beta \le -1\) is unphysical as the initial density is not locally integrable and is not treated in this work.) The case \(\beta =0\) corresponds to standard Riemann problems whose solutions are combinations of backward and forward shocks and rarefaction waves. In contrast, for \(\beta \in (-1,\infty )\smallsetminus \{0\}\), we construct the self-similar solution and show that it always contains exactly two shock waves. These are necessarily generated at time \(0+\) and move apart along straight lines. We provide a physical interpretation of the solution structure and describe the behavior of the solution in the emerging wedge between the shock waves.