Asymptotic Stability Of Periodic Solution Of One-dimensional Flowing For Van Der Waals Fluids

In this paper, we investigate the Cauchy problem of nonlinear hyperbolic-elliptic system in the Lagrangian coordinates where u(x,t) is the velocity, v(x,t)>0 is the volume, is the equation of state,μ>0 is the viscosity constant, It describes some characters of liquid-vapor phase transitions to some extent. The calculation of Van der Waals fluids'state equations is very important in the fluid mechanics, but the numerical results and experiment is always a lot of differences, mainly due to the problems in physics and mathematics on the instability caused by the ill-posedness. In this paper, we by adding artificial viscosity, so that the above equations into the elliptic mixed parabolic equations.In this paper, we add the artificial viscosity,then the system can be written as follows: The initial conditionAnd the 2L-periodic boundary conditionA model system of Van der Waals fluids has been investigated, and the asymptotic stability of solution has been established for periodic boundary value problem with artificial viscosity. A solution of steady-state periodic boundary value problem has been derived. By the means of existence and uniqueness of the local solution and the priori estimates, the steady-state solution is shown to have asymptotic stability.The main results and methods are as follows:(1) The solution's unique local existenceThrough the contraction mapping principle and the method of iterative derived the local solution of the equations existence. by the method of energy estimation obtained the local solutions of equations and uniqueness.(2) The solution's unique global existence and asymptotic propertyThe global solutions of the equations existence and uniqueness of the proof is by constructing the energy functional, combined with prior estimates obtained. Finally, the equations obtained a priori estimates and the global asymptotic properties.