On a Regularized Family of Models for Homogeneous Incompressible Two-Phase Flows

Abstract

We consider a general family of regularized models for incompressible two-phase flows based on the Allen–Cahn formulation in $$n$$ -dimensional compact Riemannian manifolds for $$n=2,3$$ . The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes- $$\alpha $$ -like model, the Leray- $$\alpha $$ model, the modified Leray- $$\alpha $$ model, the simplified Bardina model, the Navier–Stokes–Voight model, and the Navier–Stokes model) for the fluid velocity $$u$$ suitably coupled with a convective Allen–Cahn equation for the order (phase) parameter $$\phi $$ . We give a unified analysis of the entire three-parameter family of two-phase models using only abstract mapping properties of the principal dissipation and smoothing operators and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We establish existence, stability, and regularity results and some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the $$\alpha \rightarrow 0$$ limit in $$\alpha $$ models. Then we show the existence of a global attractor and exponential attractor for our general model and establish precise conditions under which each trajectory $$\left( u,\phi \right) $$ converges to a single equilibrium by means of a Lojasiewicz–Simon inequality. We also derive new results on the existence of global and exponential attractors for the regularized family of Navier–Stokes equations and magnetohydrodynamics models that improve and complement the results of Holst et al. (J Nonlinear Sci 20(5):523–567, 2010). Finally, our analysis is applied to certain regularized Ericksen–Leslie models for the hydrodynamics of liquid crystals in $$n$$ -dimensional compact Riemannian manifolds.