Unique strong and V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ \mathbb {V} }$$\end{document}-attractor of a three-dimensional globally modified two-phase flow model

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 Caraballo et al. (Adv Nonlinear Stud 6(3):411–436, 2006) for the velocity, coupled with an Allen–Cahn model for the order (phase) parameter. We prove the existence and uniqueness of strong solutions. Using the flattening property, we also prove the existence of global V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ \mathbb {V} }$$\end{document}-attractors for the model. Using a limiting argument, we derive the existence of bounded entire weak solutions for the three-dimensional coupled Allen–Cahn–Navier–Stokes system with time-independent forcing.