Cauchy Problem For A Class Of Hydrodynamic Systems

In this dissertation,we study the Cauchy problem for a class of hydrodynamic sys-tems.In Chapter 1,we give the physical background of the problems to be studied,the research status and preparatory knowledge.In Chapter 2,we study the existence of a weak solution of the Gurevich-Zybin sys-tem in the low order Sobolev space H^s（R）×H^s（R）（1<s≤₂³）by using an improved pseudo-parabolic regularization method.First,we construct the approximate system,and the local well-posedness and a prior estimates of solutions of the approximate system are given.Then by using the contracting mapping principle,the properties of autonomous ODEs in Banach space and the Aubin-Lions lemma,we obtain the existence of a weak solution.Although some important conservation laws are missing,which makes it dif-ficult to prove the global existence of approximate solutions,we solve this problem by using the structure of the system itself.In Chapter 3,we further study the existence of a weak solution to the generalized Riemann-type hydrodynamical system,which is a generalization of the Gurevich-Zybin system.By using the same method as the one applied in Chapter 2,the existence of a weak solution of the system in the low order Sobolev space（H^s（R））^N（1<s≤₂³）is obtained.In Chapter 4,the well-posedness and blow-up of a strictly hyperbolic system in a nonconservative form are studied.Firstly,the classical energy method is used to obtain local-in-time a priori estimates of solutions of the system,and then we get the existence of a solution by the standard approximation method.Secondly,the uniqueness and con-tinuous dependence on initial values of the solution are proved via the energy estimate.Finally,we establish the blow-up criterion by combining the energy estimate with the characteristic method.