Applications Of The Compensated Compactness Method On A Class Of Nonlinear Hyperbolic Systems

In this paper,we introduce some applications of the compensated compactness method to scalar conservation law as well as some important hyperbolic conservation laws.With the aid of the well-known Bernstein-Weierstrass theorem, we derive the strong convergence of a sequence of uniform L^∞or L_loc^p bounded approximate solutions for scalar equations without convexity by constructing four families of Lax entropies,and study the relaxation limit for a symmetrically hyperbolic system based on the results;by using the compensated compactness method coupled with some basic ideas of the kinetic formulation,we considerably simplify the most crucial and difficult step,reduction of Young measures,to give refined proofs for the existence of L^∞entropy solutions to a system of quadratic flux and the Le Roux system,and establish a compactness framework of theρ- u equations for the case of 1＜γ＜3.This paper consists of five chapters,the arrangement of it is as follows: In Chapter 1,we introduce some basic definitions and important theorems of the theory of compensated compactness,and prove the existence of smooth solutions for the parabolic system.In Chapter 2,we derive the strong convergence of a sequence of uniform L^∞or L_loc^P bounded approximate solutions for scalar equations without convexity by constructing four families of Lax entropies,and study the relaxation limit lot a symmetrically hyperbolic system based on the results.In Chapter 3,4,we use the compensated compactness method coupled with some basic ideas of the kinetic formulation to simplify the proofs for the existence of globally bounded entropy solutions to a system of quadratic flux and the Le Roux system,and then obtain some more general results.The zero relaxation limits for the two systems are also discussed. In Chapter 5,we establish a compactness framework of the Euler equations of one dimensional,compressible fluid flow(ρ- u equations)for the case of 1＜γ＜3,and give a much concise proof for the existence of entropy solutions to one dimensional isentropic gas dynamics(ρ- m equations)with 1＜γ＜3,provided that the initial data are away from vacuum state.We also obtain global existence for some importantρ- u equations with sources as well asρ- m equations with sources.