The Study Of Solutions For The Fluid Equations With Korteweg Term

In this paper,our research focus on two respects mainly.On the one hand,we are concerned with the asymptotic behavior of solutions to Euler-Korteweg equations with damping.We prove that the solutions time-asymptotically behave like the nonlinear diffusion wave and get the corresponding convergence rate.Moreover,it is also shown that the Euler-Korteweg equations with damping could be a approximation of the Euler equations with damping.On the other hand,our research will focus on the global existence of weak solutions to one-dimensional liquid-gas model with Korteweg term.In the first part of this paper,we can get the expression of correction functions based on the property of Euler-Korteweg equations with damping and Darcy laws in the process of proving.So we can translate the Euler-Korteweg equation with damping into a new equation.Then we construct the self-similar solutions to the nonlinear diffusion equation to obtain dissipative estimation of txv),(.At last,we will study the decay rate of the new equation rather than study convergence to nonlinear diffusion waves for solutions of Euler-Korteweg equations with damping.In the second part of this paper,we will use the method of energy estimation to prove the global existence of weak solutions for the two-phase flow model with Korteweg term.