Global Existence Of Solutions For Quasilinear Hyperbolic Systems Of Balance Laws

This thesis concerns with the global existence of smooth solu-tions of quasilinear hyperbolic systems of balance laws. The thesis isorganized as follows.Chapter 1 is an introduction. In this chapter, we simply recallthe physical background and present situation of the study on theCauchy problem and Riemann problem of quasilinear hyperbolic sys-tems of first order. We illustrate the problems which we shall discussand state the main results contained.In Chapter 2, we consider a kind of quasilinear hyperbolic sys-tems with inhomogeneous terms satisfying dissipative condition ormatching condition. For the Cauchy problem of this kind of systems,we prove that, if the system is weakly linearly degenerate and theinitial data is small and satisfies some decay condition, then theCauchy problem admits a unique global classical solution on t≥0.In Chapter 3, we investigate a class of mixed initial-boundaryvalue problems for a kind of n×n quasilinear hyperbolic systems ofconservation laws on the quarter plan. We show that the structure of the piecewise C~1 solution u = u(t,x) of the problem, which can beregarded as a perturbation of the corresponding Riemann problem, isglobally similar to that of the solution u = U(z/t) of the correspondingRiemann problem. The piecewise C~1 solution u = u(t,x) to this kindof problems is globally structure stable if and only if it contains onlynon-degenerate shocks and contact discontinuities, but no rarefactionwaves and other weak discontinuities.