Global Existence And Asymptotic Behavior Of Solutions For The One-dimensional Compressible Micropolar Fluid Model With Large Initial Data

The compressible micropolar fluid model describes the motions of a variety of complex fluids consisting of dipole elements such as the suspensions,animal blood,liquid crystal,etc.The mathematical theory of this model is one of the hottest topics in the area of partial differential equations in the last several decades.Up to now,although the results for the small initial data are almost complete,the results for the large initial data are far from complete.Especially,few results have been obtained for the global large solutions to the Cauchy problem of this model.This thesis is mainly concerned with the global existence and large-time behavior of strong solutions with large initial data for the following Cauchy problem of the one-dimensional isentropic compressible fluid model with density-dependent viscosity and microviscosity coefficient:(?) Here t and x denote the time variable and the spatial variable in the Lagrangian coordinates,respectively.The unknown functions v(t,x),u(t,x),ω(t,x)and p(v)are the specific volume,the velocity,the microrotation velocity and the pressure of the fluid,respectively.μ(v)is the viscosity coefficient,A(v),μr(v)are the coefficients of microviscosity.Furthermore,v±≥0,u± and ω± are given constants.Throughout this paper,we assume that the pressure p(v)and the viscosity coefficient μ(v)are given by p(v)=u-γ,μ(v)=v-α,γ≥1,α≥0,(2)where the microviscosity coefficients A(v)and μr(v)are smooth positive functions of v>0,γ≥1 denotes the adiabatic exponent of gas,and α≥0 is a parameter.The first part of this thesis is concerned with the case when the far fields of the initial data are different,i.e.,(v+,u+,ω+)≠(v-,u-,ω-).In this case,we consider the global nonlinear stability of rarefaction waves for the Cauchy problem(1).Here global stability means the initial perturbation can be arbitrarily large.By using the elementary energy method and Kanel’s technique,we prove that if the parameters α and γ satisfy some conditions,and the initial data is sufficiently regular,without vacuum and mass concentrations,then the Cauchy problem(1)has a unique global strong nonvacuum solution,which tends to a superposition of two rarefaction waves of the corresponding Euler equations as time goes to infinity.This result holds for arbitrarily large initial perturbation and large-amplitudes rarefaction waves.Moreover,the exponential time decay rate of the microrotation velocity ω(t,x)under large initial perturbation is also derived by employing the time weighted energy method.In the second part of this thesis,we consider the case when the far fields of the initial data are the same,i.e.,(v+,v+,ω+)=(v-,u-,ω-)=(v,u,0).By using the same method as the first part,we obtain the global existence and large-time behavior of strong solutions with large initial data for the Cauchy problem(1)under the perturbation of the constant state(v,u,0).Furthermore,by using the time weighted energy method,the algebraic time decay rates for the specific volume v(t,x)and the velocity u(t,x)are also established under partially large initial perturbation.This thesis is divided into four chapters.In the first chapter,we introduce our problem and some related background,and then state the three main theorems of this thesis.In the second chapter,we shall prove the global stability of rarefaction waves.To do so,we first give the properties of the smooth rarefaction waves,and then we are concerned with the global stability of rarefaction waves by the properties of the smooth rarefaction waves,the elementary energy method and Kanel’s technique.Moreover,the exponential time decay rate of the microrotation velocity ω(t,x)is also derived.Chapter three is devoted to proving the global existence and large-time behavior of strong solutions with large initial data for the Cauchy problem(1)under the perturbation of the constant state.We also deduce the algebraic decay rate of specific volnmev(t,x)and the velocity u(t,x)in this chapter.In Chapter four,we summarize the whole thesis,and then present some problems which deserve for further investigation.