Global Weakly Discontinuous Solutions For First Order Quasilinear Hyperbolic Systems

In this Ph.D. thesis, we study systematically the weakly discontinuous solutions for first order quasilinear hyperbolic systems. The existence and uniqueness of global weakly discontinuous solution are obtained for Cauchy problem and mixed initial-boundary value problem respectively.The arrangement of the thesis is as follows:First of all, a brief introduction on the study of classical solutions for first order quasilinear hyperbolic systems and the main results of this thesis are given in chapter 1．For convenience, in chapter 2, we list some preliminaries, including some definitions, such as normalized coordinates, generalized normalized coordinates, weak linear degener-acy and weakly discontinuous solution, and the John's formula on the decomposition of waves.In chapter 3, we study the Cauchy problem with a kind of non-smooth initial data for general quasilinear hyperbolic systems with characteristics with constant multiplicity. A necessary and sufficient condition is given to guarantee the existence and uniqueness of global weakly discontinuous solution for the Cauchy problem. The main result obtained can be used to the general system of the motion of an elastic string and the time-like extremal surface in Minkowski space R^1+(n+1) .We consider the Cauchy problem with a kind of non-smooth initial data for first order inhomogeneous quasilinear hyperbolic systems in chapter 4．Under the assumption that the inhomogeneous term satisfies the matching condition, a necessary and sufficient condition for the global existence of weakly discontinuous solution for the Cauchy problem is obtained.Chapter 5 and 6 are devoted to the study of the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems.In chapter 5, we consider the mixed initial-boundary value problem for homogeneous quasilineai hyperbolic systems. If the initial and boundary data satisfy some "small and decaying " condition, it is proved that this problem admits a unique global weakly discon-tinuous solution. The main result is then applied to the mixed problem for system of the motion of an elastic string.Chapter 6 is the continuation of the former chapter. If the inhomogeneous term satisfies the matching condition, by proving a lemma to simplify the formulas on the decomposition of waves, we get the existence of global weakly discontinuous solution to the mixed initial-boundary value problem for inhomogeneous quasilinear hyperbolic systems.