Decay Property Of Solution To The Euler-Cattaneo-Maxwell System

This thesis is devoted to the decay problem of Euler-Cattaneo-Maxwell equations in the Sobolev space.In the literature[12],Kawashima and Ueda proved the global-in-time existence of solutions.Also,the decay property for the corresponding linearized equations is also investigated.Furthermore,in the thesis,we show that solutions to the nonlinear equations decay at the rate t^-3/4 when the time t?? by using weighted energy methods,provided that the initial date are in H^s???L¹ sufficiently small and the regularity satisfies s?6.The paper is divided into the following four chapters.In the first chapter,we introduce the Euler-Cattaneo-Maxwell model and analyze its physical background.Then,we briefly state the research development for isentropic Euler-Maxwell equations.The research motivation,major difficulty and the corresponding strategy are also given.Finally,main results of this paper are stated.In the second chapter,for convenience of the proof,we introduce Sobolev spaces and present briefly the space definition and basic properties.In the third chapter,we begin to prove main theorems of this thesis.Firstly,by using the time-weighted energy method,we established a nonlinear energy inequality,which leads to the time decay of solutions.Secondly,based on the decay property of solutions to linearized equations,the optimal decay rates of solutions are obtained with aid of the Duhamel principle.In the fourth chapter,we give a summary in this thesis and some questions in near future.