Global solutions with shock waves to the generalized Riemann problem for a system of hyperbolic conservation laws with linear damping

Abstract

It is proven that a class of the generalized Riemann problem for quasilinear hyperbolic systems of conservation laws with the uniform damping term admits a unique global piecewise C 1 solution u = u ( t , x ) containing only n shock waves with small amplitude on t ⩾ 0 and this solution possesses a global structure similar to that of the similarity solution u = U ( x t ) of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data. We also give an example to show that the uniform damping mechanism is not strong enough to prevent the formation of shock waves.