Qualitative Analysis Of The Solutions For Two Kinds Of Nonlinear Hyperbolic Burgers Equations

This paper mainly studies the properties of solutions for two kinds of nonlinear hyperbolic Burgers equations in Sobolev spaces.Firstly,we study the local well-posedness,blow up criteria and continuity for the 1D Euler-Alignment system,which comes from in the field of complex biological systems simulating animal population migration.Secondly,we investigate the existence and uniqueness of weak solutions and blow up criteria of strong solutions for a model(classical Boussinesq system),which describes the propagation of small amplitude long waves on the surface of an ideal fluid in a uniform horizontal channel with finite depth.This main contents are as follows:The first part includes the second chapter and the third chapter,which mainly studies the local well-posedness,blow up criteria and continuity for the 1D Euler-Alignment system.Firstly,the local well-posedness of the 1D Euler-Alignment system in Sobolev spaces H^s+1(R)×H^s(R)(s>1/2)is established by using Littlewood-Paley decomposition and the transport equation theory,and the criteria of blow up of the 1D Euler-Alignment system in Sobolev spaces is obtained by using the method of energy estimation.The local well-posedness shows that the mapping from initial value to solution of the 1D EulerAlignment system is continuous.Further,we consider that the mapping is not uniformly continuous.That is to construct two sequences of solution for the 1D Euler-Alignment system in Sobolev spaces H^s+1(R)× H^s(R)(s>1/2).The distance of the solution sequences converges to zero when t=0,but there is a lower bound when any t>0,and the lower bound is strictly greater than zero,which indicates that the mapping from initial value to solution is not uniformly continuous.Furthermore,it is obtained that the solution mapping of the 1D Euler-Alignment system is Holder continuous in H^?+1(R)×H^?(R) topology for all-1/2<?<s.In the second part,we mainly investigate the existence and uniqueness of weak solutions and blow up criteria of strong solutions for the classical Boussinesq systems.Firstly,we establish the local well-posedness of the classical Boussinesq system in Sobolev spaces H^s(R)× H^s-1(R)(s>3/2)by using Kato's theorem.Then,the existence and uniqueness of weak solution of the classical Boussinesq system in Sobolev spaces H^s(R)×H^s-1(R)(s?1)is established by using the quasi parabolic regularization method.Finally,by using the transport equation theory and Sobolev inequality,the sufficient conditions for the global strong solution are obtained.Then,by using the sufficient conditions and energy estimation,we obtain the blow up criterion of the strong solution of the classical Boussinesq system in Sobolev spaces H^s(R)×H^s-1(R)(s>3/2),that is,the blow up occurs for this system only in the form of breaking waves.