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

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Abstract

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 Ul= Ur. (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.

Original languageEnglish (US)
Article number61
JournalZeitschrift fur Angewandte Mathematik und Physik
Volume72
Issue number2
DOIs
StatePublished - Apr 2021

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Physics and Astronomy(all)
  • Applied Mathematics

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