On the Solution to the Riemann Problem for Arbitrary Hyperbolic System of Conservation Laws

Using the evolutionary condition proposed by the author in ref. [11] we prove certain theorems related to the uniqueness of self-similar solutions to the Riemann problem for arbitrary strictly hyperbolic system of quasilinear conservation laws. The solutions are sought in the class of piecewise smooth vector functions of two variables and are assumed to satisfy the equations and the initial and boundary data in the weak sense. The discontinuity waves considered here include shock and reaction waves. We also prove dual extremum principles which are a partial extension of the Bethe-Weyl theory of hydrodynamic shock waves. The existence and certain global properties of Hugoniots are established. The case of degenerate eigenvalues (nonstrict hyperbolicity) is also considered. Applications to hydrodynamics, detonation and deflagration waves, shear waves, nonlinear thermoelastic plane waves follow.

CONTENTS

1. Introduction, ...5
2. The evolutionary condition and the solutions to the Riemann problem, ...8
3. Sonic discontinuity waves, ...16
4. The groups of k-waves in a neighborhood of a sonic point, ...30
5. Continuation of groups k-waves beyond critical points, ...39
6. A dual extremum theorem for Chapman-Jouguet points, ...43
7. Existence and asymptotic properties of Hugoniots, ...47
8. Applications to hydrodynamics; detonation and deflagration waves, ...60
9. Plane waves in compressible thermoelastic media, ...77
10. Plane waves in thermoelastic Hadamard materials, ...100
11. Plane waves in incompressible thermoelastic materials, ...103
12. Dissipation associated with solutions of hyperbolic systems of conservation laws, ...113
13. Remarks, ...118
References, ...119
Polish summary (Streszczenie), ...122