Stability Of Phase Transition For Compressible Van Der Waals Fluids

To analyze the flow of the non-ideal fluids is an important problem in the field of partial differential equations. In Euler coordinate, the flow of one-dimensional non-ideal fluid can be described the following nonlinear partial differential equations Here (x,t)∈R×R+,R=(-∞,+∞)，R+=(0,+∞), u(x,t) is the velocity, p(x,t)>0the density, μ>0the viscous constant, p=p(p) the stress function, which satisfies the function of Van der Waals as follows: p(ρ=-3ρ2=(?)+(8(?)ρ)/(3-ρ) Here (?) is the ratio of absolute temperature and critical temperature. That is the non-ideal fluids when (?) is positive. This paper mainly researched the non-ideal fluid.Considering the equations (1)-(2) ill-posed in the mathematics, we usually introduce the artificial viscosity to overcome this difficulty. In recent years, depending on the Lagrange transformation, people obtained the fluids’phase transition and the relationship of flow of the fluids between the initial density and velocity. However, the initial boundary value problem is not able to take the method of Lagrange transformation so that it has not done any particularly meaningful results up to the present.In this paper, to get the method of energy estimation discussed the existence of global solution and the large time behavior of the periodic boundary value problem of the equations (1)-(2) in Euler coordinate. And we proved that the flow’s solutions of the fluids converge to the steady-state solutions under small disturbance of the initial density and momentum. The results accord with the fact.