Global weakly discontinuous solutions to quasilinear hyperbolic systems of conservation laws with damping with a kind of non-smooth initial data

Abstract

For the Cauchy problem with a kind of non-smooth initial data for weakly linearly degenerate hyperbolic systems of conservation laws with the linear damping term, we prove the existence and uniqueness of global weakly discontinuous solution u = u(t, x) containing only n weak discontinuities with small amplitude on t ≥ 0, and this solution possesses a global structure similar to that of the similarity solution $$u = U(\frac{x}{t})$$ of the corresponding homogeneous Riemann problem. As an application of our result, we obtain the existence and uniqueness of global weakly discontinuous solution, continuous and piecewise C 1 solution with discontinuous first order derivatives, of the flow equations of a model class of fluids with viscosity induced by fading memory.