Global Classical Solutions To The Compressible Navier-Stokes-Korteweg Equations

This thesis is concerned with the isothermal compressible Navier-Stokes-Korteweg equations and nonisothermal compressible Navier-Stokes-Korteweg equations. Navier-Stokes-Korteweg system is one of the model describing the motion of viscous fluid substance. Due to the complexity of the phenomena described by the compressible Navier-Stokes-Korteweg equations and the high nonlinearities of the equations, the compressible Navier-Stokes-Korteweg equations becomes one of the active reach in mathematics. The thesis consists of four chapters:In Chapter One, first, we summarize the background of the related problems. Second, we state the main results of the present thesis. Last,we also give some preliminary results and notations used in the whole thesis.In Chapter Two, we study the Cauchy problem of the isothermal compressible Navier-Stokes-Korteweg equations. Precisely, we give the global classical solutions under the condition that the initial energy is small enough.In Chapter Three, we investigate the Cauchy problem of the nonisothermal com-pressible Navier-Stokes-Korteweg equations. Similarly, we obtain the global classical solutions if the initial energy is small enough.In Chapter Four, we consider the nonisothermal compressible Navier-Stokes-Korteweg equations when the capillary coefficient κâ†'0. When the capillary co-efficient vanishes, the solution of the problem converges to the corresponding solu-tion of the compressible Navier-Stokes equations Solution; Also, the convergence rate estimates about the capillary coefficient κ are given.