In this article we study a globally modified Allen–Cahn–Navier–Stokes system in a three-dimensional domain. The model consists of the globally modified Navier–Stokes equations proposed in (Adv. Nonlinear Stud. 6 (2006) 411–436) for the velocity, coupled with an Allen–Cahn model for the order (phase) parameter. We discretize these equations in time using the implicit Euler scheme and we prove that the approximate solution is uniformly bounded. We also show that the sequence of the approximate solutions of the globally modified Allen–Cahn–Navier–Stokes system converges, as the parameter N goes to infinity, to the solution of the corresponding discrete two-phase flow system. Using the uniform stability of the scheme and the theory of the multi-valued attractors, we then prove that the discrete attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.

中文翻译：

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三维全局修正两相流模型的隐式Euler格式的长期稳定性

在本文中，我们研究了在三维域中全局修改的Allen-Cahn-Navier-Stokes系统。该模型由（Adv。Nonlinear Stud。6（2006）411-436）中提出的针对速度的全局修改的Navier-Stokes方程，以及用于阶数（相位）参数的Allen-Cahn模型组成。我们使用隐式Euler方案及时离散这些方程，并证明了近似解是一致有界的。我们还表明，随着参数N趋于无穷大，全局修改的Allen-Cahn-Navier-Stokes系统的近似解的序列收敛到相应的离散两相流系统的解。利用该方案的一致稳定性和多值吸引子的理论，