Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws

It is proven that the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws admits a unique global piecewise C1C1 solution u=u(t,x)u=u(t,x) containing only nn shock waves with small amplitude on t⩾0t⩾0 and this solution possesses a global structure similar to that of the similarity solution u=U(x/t)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.