Global Non-linear Stability Of Some Basic Waves For The One-dimensional Compressible Navier-Stokes-Korteweg Equations

This paper is concerned with the large-time behavior of solutions to the following Cauchy problem of the one-dimensional compressible Navier-Stokes-Korteweg equations:(?)Here x?R?t?0,the unknown functions are the the specific volume v(x,t)>0,the velocity u(x,t),the temperature?(x,t)>0,and the pres-sure p(x,t)of the fluid,respectively,while ?(v,?),?(v,?),?(v,?)denote the viscosity coefficient,the capillary coefficient and the heat conductive coeffi-cient respectively.Cv>0,v±>0,u±and?±>0 are given constants,and we assume(v0,u0,?0)(±?)=(v±,u±,?±)as compatibility conditions.The Korteweg stress tensor K and the nonlinear terms F are given by(?) Throughout this paper,we suppose that the pressure P(v,?)and the constant Cv are given by(?) where s is the entropy of the fluid and ?>1,A and R are positive constants.Under some smallness assumptions on ?—1 and the capillarity coefficient?(v,?),we prove the global stability of the single viscous contact wave as well as the combination of the viscous contact wave with two rarefaction waves for the Cauchy problem(1)by using the elementary energy method combined with the technique of Y.Kanel.Here global stability means the initial perturbation can be arbitrarily large.The key ingredient in the proof is to derive the uniform positive lower and upper bounds on the specific volume v and the temperature?.This paper is divided into five chapters.In the first chapter,we mainly introduces the problems under consideration and the background.Two main theorems of this paper will be stated in this chapter.In the second chapter,we will give some important lemmas for later use.In the third chapter,we shall prove our first main theorem 1.1,i.e.,the global stability of the viscous contact wave for Cauchy problem(1).To do so,we first make the a priori assumption(?)for some positive constant N1,then the lower and upper bounds of temperature can be obtained by using the Sobolev inequality.The lower and upper bounds of specific volume v(t,x)are achieved by using the techniques developed by Y.Kanel.The fourth chapter is devoted to proving the second main theorem 1.2,i.e.,the global stability of the combination of the viscous contact wave and two rarefaction waves for Cauchy problem(1).The proof of which is similar to that of Theorem 1.1,but with an additional difficulty to control the interactions of wave from different families.In the fifth chapter,we give a summary of this thesis,and some valueable questions for further study.