The Global Smooth Solution Of A Class Of Hyperbolic Conservation Laws

This paper is concerned with the existence of global smooth solution to the Cauchy problem and the initial boundary value problem for a class of quasilinear hyperbolic systems, which are very important on applications. The whole paper involves two parts:In the first part, we consider the initial boundary value problem for a viscoelastic model with relaxationwith initial dataand boundary conditionwhere δ> 0, 0 < μ< 1, u ∈R are constants.According to particular form of (0.1.1), we overcome the difficulties caused by boundary effect by employing some function transformations and the " maximum principle " of first-order quasilinear hyperbolic system, then obtain the existence of global solution to the initial boundary value problem (0.1.1) - (0.1.3).In the second part, we consider the Cauchy problem for the one-dimensional rclativis-tic Euler equationswith initial datawhere c is a sufficiently large positive constant, the speed of light, and P is a given function of ρ.Under the assumption of the initial data ρ0(X),u0(X) which are smooth and some monotonous properties(see(A2)), we prove that the Cauchy problem (0.1.4), (0.1.5) admits a unique global smooth solution by using the similar approach used in the first part.Keywords:Viscoelastic model,relaxation,a priori estimates,maximum principle,relativistic Euler equations,global smooth solution...