The Asymptotic Behavior Of Solutions For The 1-D Compressible Micropolar Fluid Model

This paper is concerned with the asymptotic behavior of solutions to the one-dimensional compressible micropolax fluid model.Up to now,there are many results on the compressible micropolar fluid model with viscosity,but few results have been obtained on the compressible micropolax fluid model without viscosity.The first part of this paper is devoted to studying the asymptotic behavior of solutions to the one-dimensional compressible microp-olar fluid model without viscosity,i.e.,we will study the nonlinear stability of a composite wave for the following one-dimensional compressible micropolar fluid model without viscosity in the Lagrangian coordinates:ith the initial and far field conditions:ere the unknown functions v(x,t)>,0(x,t),(x,t)>0,?(x,t),e and p denote the specific volume,the velocity,the absolute temperature,the mi-rorotation velocity the internal energy and the pressure p(v,?)of the fluid,espectively,? and A are the heat conductivity coeeffcient and the mcroovs-cosity coeeffcient,respectively.v±>0,u±,?±>0,?± are given constants.We assume that(v0,u0,?0,?0)(±?)=(v±,u±,?±,?±)as compatibility condition-s.Furthermore,we assume that the pressure p(v,?)and the internal energy e(t,x)are given by:p(v,?)= R?/v=Bv-?exp(?-1/Rs),e=R/?-1?,where s is the entropy of the fluid,and ?>1,B and R are positive constants.By using the elementary energy method and the properties of the viscous contact wave and the smooth approximate rarefaction waves,we prove the nonlinear stability of the single viscous contact wave and a composite wave consisting the superposition of the viscous contact wave and two rarefaction waves for the Cauchy problem(1)-(2),provided that the initial data and the strength of the waves are sufficiently small.Secondly,there are many results on the compressible micropolar flow mod-el with small initial data,but fewer results have been obtained on this model with large initial data.In the second part of this paper,we will consider the global existence and large-time behavior of smooth large solutions to the following one-dimensional isentropic compressible micropolar fluid model:with the initial and far field conditions:Here ?(v)and A(v)denote the viscosity coefficient and the microviscosity coefficient respectively;? and ? are constants;the physical meaning of other qualities are the same as those in(1).By using the elementary energy method together with the technique of Y.Kanel,we obtain the global existence and large-time behavior of smooth solutions around constant state to the Cauchy problem(3)-(4)with large initial data,and also the global stability of rarefaction waves for the the Cauchy problem(3)-(4).The key point in the proof is to derive the uniform-in-time lower and upper bounds for the specific volume v(t,x).This thesis is divided into five chapters.In the first chapter,we introduce our problem and some related background,and then state the four main the-orems in the thesis.In the second chapter,we give some important lemmas which will be used in the proof of our main theorems.Chapter three is devoted to proving the main Theorems 1.1 and 1.2.In these two theorems,by the ele-mentary energy method,we obtain the nonlinear of the single viscous contact wave and the nonlinear stability of a composite wave consisting the superpo-sition of the viscous contact wave and two rarefaction waves for the Cauchy problem(1)-(2)under small initial perturbations,respectively.Since the proof of Theorem 1.1 is similar to that of Theorem 1.2,we only present here the detailed proof of Theorem 1.2(i.e.,the nonlinear stability of composite wave).In the fourth chapter,we prove Theorem 1.3,i.e,,the global existence and large-time behavior of smooth solutions around constant state to the Cauchy problem(3)-(4)with large initial data,and Theorem 1.4,i.e.,the global stabil-ity of rarefaction waves for the the Cauchy problem(3)-(4).To do so,we first use the technique of Y.Kanel to derive the uniform-in-time lower and upper bounds for the specific volume v(t,x),and then obtain Theorem 1.3 and 1.4 by using the method of energy estimates and some standard techniques.In the last chapter,we summarize the whole thesis,and then present some problems that deserve further study in the future.