Shock reflection for general quasilinear hyperbolic systems of conservation laws

Abstract

This paper concerns the reflection of shock waves for general quasilinear hyperbolic systems of conservation laws in one space dimension. It is shown that the mixed initial–boundary value problem for general quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions on the quarter-plane { ( t , x ) ∣ t ≥ 0 , x ≥ 0 } admits a unique global piecewise C 1 solution u = u ( t , x ) containing only shock waves with small amplitude and this solution possesses a global structure similar to that of the Riemann solution u = U ( x t ) of the corresponding Riemann problem, if the positive eigenvalues are genuinely nonlinear and the Riemann solution has only shock waves, and no rarefaction waves and contact discontinuities. Our result indicates that the Riemann solution u = U ( x t ) consisting of only shock waves possesses a semi-global nonlinear structure stability.