Mechanism Of The Formation Of Singularities For First Order Quasilinear Hyperbolic Systems

In this Ph.D. thesis, we consider the mechanism of the formation of singularities for first order quasilinear hyperbolic systems.First of all, a brief introduction of the background and the present situation on the study of mechanism of the formation of singularities for first order quasilinear hyperbolic systems is given in Chapter 1.For convenience, in Chapter 2, we list some preliminaries, including several definitions, such as normalized coordinates, weak linear degeneracy, weakly discontinuous solution, and the John's formula on the decomposition of waves.One often believes that there is no shock formation for the Cauchy problem of quasilinear hyperbolic systems (of conservation laws) with linearly degenerate characteristic fields. It has been a conjecture for a long time (see [4], [42]) and it is still an open problem in the general situation up to now. In Chapter 3, a framework to justify this conjecture is proposed, and, by means of the concept such as the strict block-hyperbolicity, the part richness and the successively block-closed system, some general kinds of quasilinear hyperbolic systems, which verify the conjecture, are presented.In Chapter 4, we consider the mixed initial-boundary value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant {(t, x)| t≥0, x≥0}. Under the assumptions that the system is strictly hyperbolic and linearly degenerate or weakly linearly degenerate, the global existence, uniqueness and L~1 stability of C~1 solutions are obtained for small initial and boundary data. We also give two applications for corresponding physical models.In the last chapter, for inhomogeneous diagonal system with distinct characteristics or with characteristics with constant multiplicity, under the assumption that the system is linearly degenerate and the C~1 norm of the initial data is bounded, we show that the mechanism of the formation of singularities of classical solution to its Cauchy problem must be of ODE type. Similar results are also obtained for corresponding mixed initial-boundary value problems on a semi-unbounded domain, Goursat problem with C~1 boundary conditions or with weakly discontinuous boundary conditions.