The Global Existence And Singularities Of Smooth Solutions For Hyperbolic Conservation Laws

In this paper, we mainly focus on the global existence and blowup of smooth solution for the Cauchy problem of hyperbolic systems in one and several space variables.In Chapter1, we introduce the background of our study and the results obtained by others in hyperbolic systems and the main results of our research papers.In Chapter2, we study a class of complex conservation laws, which is intro-duced by P. D. Lax and essentially a class of quasilinear hyperbolic systems in two space dimensions. For a kind of flux functions, the system is linearly degen-erate in the sense of P. D. Lax, in this case, we prove the global existence and lower bound of the lifespan of smooth solutions. For other kind of flux functions, the system is genuinely nonlinear in some directions on characteristic fields, in this case, we prove that the smooth solution of the Cauchy problem must blowup in finite time and give a sharp estimate on the lifespan of the smooth solution.Chapter3is on the formation and propagation of singularities in one dimen-sional Chaplygin gas. Under suitable assumptions on the initial data, we obtain a new type of singularities (Delta-like singularity), including "point-like ","line-like" and "cusp point-like". By the method of characteristics, we find out that this kind of singularity is different from the shock waves and the mechanism of the formation of this kind of singularity is due to the envelope of different fam-ilies of characteristics (i.e., the loss of strictly hyperbolic). Besides, we analyse the behavior of the solution in a neighborhood of a blowup point. Furthermore, by the behavior of the solution derived above, we can construct a unique weak solution (δ-shock) after blowing up of the solution.In Chapter4, we study the generalized timelike extremal surface equation in de Sitter spacetime, which plays an important role in both mathematics and physics. Under the assumption of small initial data with compact support, we investigate the lower bound of lifespan of smooth solutions by weighted energy estimates.Finally, in Chapter5, we introduce a concept of "completely linearly de-generate", which generalizes the concept of "linearly degenerate" in the sense of P. D. Lax. Under the definition of "completely linearly degenerate", we get an interesting phenomenon, which says that for a symmetric hyperbolic conserva-tion laws with two unknowns,"completely linearly degenerate" is equivalent to "linearity"...