The generalized Riemann problem for first order quasilinear hyperbolic systems of conservation laws I

摘要

In this paper, we consider a generalized Riemann problem of the first order hyperbolic conservation laws. For the case that excludes the centered wave, we prove that the generalized Riemann problem admits a unique piecewise smooth solution $u=u(t,x)$, and this solution has a structure similar to the similarity solution $u=Uleft(rac{x}{t}ight)$ of the corresponding Riemann problem in the neighborhood of the origin provided that the coefficients of the system and the initial conditions are sufficiently smooth.