一维Navier-Stokes-Cahn-Hilliard方程组解的适定性分析

讨论和描述了具有扩散界面的互不相溶气液两相流动的可压缩Navier-Stokes-Cahn-Hilliard（NSCH）方程组的周期边值问题，NSCH方程组中采用了van der Waals状态方程，该状态方程是关于密度非凸的刻画气液相变的经典模型。通过对压力的单调分解并结合能量估计的方法，克服了状态方程非凸性带来的困难，得到了流体密度的上下界估计；对任意初始值（密度不含真空），证明了该问题的一维流动强解是全局存在且唯一的。结果表明，该气液相变问题不会出现激波和真空现象。

Well-posedness of solutions for Navier-Stokes-Cahn-Hilliard system in one dimension

This paper is concerned with a diffusive interface model for immiscible gas-liquid binary fluids. The periodic boundary value problem for a compressible Navier-Stokes-Cahn-Hilliard equation is discussed, and the van der Waals equation of state, which is non-convex for density and is the classical model for gas-liquid phase transition, is employed. In addition, we make use of the monotonic decomposition of pressure combined with the energy estimation method to overcome the difficulty caused by the non-convexity of the state equation, and the upper and lower bounds of the density are obtained. For any initial value (density without vacuum), the global existence and uniqueness of strong solutions is thus proved. The results indicate that there is no shock or vacuum phenomenon in the gas-liquid phase change problem.