Conjecture on the structure of solutions of the Riemann problem for two- dimensional gas dynamics systems. (English) Zbl 0726.35081

Zhang, Tong; Zheng, Yuxi

SIAM J. Math. Anal. 21, No. 3, 593-630 (1990).

The authors consider the Riemann problem for the gas dynamics systems of isentropic and adiabatic flow, in two dimensions of space. For instance, in the first case the system is the familiar \[ \rho_ t+(\rho u)_ x+(\rho v)_ y=0,\quad (\rho u)_ t+(\rho u^ 2+p)_ x+(\rho uv)_ y=0,\quad (\rho v)_ t+(\rho uv)_ x+(\rho v^ 2+p)_ y=0, \] where \(\rho\) is the density, (u,v) the velocity, \(p=A\rho^{\gamma}\), \(\gamma >1\), \(A>0\) the pressure. The initial data are constant in each quadrant of (x,y) plane; each jump can be connected by exactly one planar rarefaction or shock wave, or by a slip line. The solutions are assumed to depend only on \(\xi =x/t\), \(\eta =y/t\); with such a change of coordinates the Riemann problem reduces to a boundary value problem at infinity, and one looks for global solutions in (\(\xi\),\(\eta\)) plane.
Extending the planar flow from infinity along the stream lines, one has to solve many problems connected to the interactions of these waves: global solutions for some Goursat problems, degenerate elliptic systems with free boundaries, reflection problems. If all these problems are solvable (these are the conjectures) then the solutions consist of constant states, rarefaction or shock waves and slip lines. The slip lines can even form spirals, and other interesting features (shocks with bifurcation points, or vanishing continuously) appear. Many figures for the different situations are shown.

Reviewer: A.Corli (Ferrara)

MSC:

35L65	Hyperbolic conservation laws
35L67	Shocks and singularities for hyperbolic equations
65M99	Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
76P05	Rarefied gas flows, Boltzmann equation in fluid mechanics

Keywords:

rarefaction waves; slip plane; selfsimilar solutions; rarefactive pseudostationary flow; Riemann problem; gas dynamics; shock waves