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

We consider self-similar solutions to the 1-dimensional isothermal Euler system for compressible gas dynamics. For eachβ∈ R, the system admits solutions of the form ρ(t,x)=tβΩ(ξ)u(t,x)=U(ξ)ξ=xt,where ρ and u denote the density and velocity fields. The ODEs for Ω and U can be solved implicitly and yield the solution to generalized Riemann problems with initial data ρ(0,x)={Rl|x|βx<0Rrxβx>0u(0,x)={Ulx<0Urx>0,where Rl,Rr>0 and Ul,Ur are arbitrary constants. For β∈ (- 1 , 0) , the data are locally integrable but unbounded at x= 0 , while for β∈ (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 β> 1 sufficiently large and U_l= U_r. (The case β≤ - 1 is unphysical as the initial density is not locally integrable and is not treated in this work.) The case β= 0 corresponds to standard Riemann problems whose solutions are combinations of backward and forward shocks and rarefaction waves. In contrast, for β∈ (- 1 , ∞) \ { 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.