Well-posedness of three-dimensional damped Cahn–Hilliard–Navier–Stokes equations

This paper presents a mathematical analysis of the evolution of a mixture of two incompressible, isothermal fluids flowing through a porous medium in a three-dimensional bounded domain. The model is governed by a coupled system of convective Brinkman–Forchheimer equations and the Cahn–Hilliard equation, considering a regular potential and nondegenerate mobility. We first establish the existence of a Leray–Hopf weak solution for the coupled system when the absorption exponent [Formula: see text]. Additionally, we prove that every weak solution satisfies the energy equality for [Formula: see text]. This further leads to the uniqueness of weak solutions in three-dimensional bounded domains, subject to certain restrictions on the viscosity ([Formula: see text]) and the Forchheimer coefficient ([Formula: see text]) in the critical case [Formula: see text]. Similar results are also obtained for the case of degenerate mobility and singular potential.