Well-posedness of three-dimensional Damped Cahn-Hilliard-Navier-Stokes Equations

TL;DR

This paper rigorously analyzes the well-posedness of a complex coupled PDE system modeling two-fluid flow in porous media, establishing existence, energy equality, and uniqueness of solutions under various conditions.

Contribution

It provides the first comprehensive proof of existence and uniqueness of weak solutions for the three-dimensional damped Cahn-Hilliard-Navier-Stokes system with regular and degenerate potentials.

Findings

Existence of Leray Hopf weak solutions for r ≥ 1

Energy equality holds for solutions with r > 3

Uniqueness of solutions under certain viscosity and coefficient restrictions

Abstract

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 non degenerate mobility. We first establish the existence of a Leray Hopf weak solution for the coupled system when the absorption exponent r greater than or equal to 1. Additionally, we prove that every weak solution satisfies the energy equality for greater than 3. This further leads to the uniqueness of weak solutions in three-dimensional bounded domains, subject to certain restrictions on the viscosity and the Forchheimer coefficient in the critical case r=3. Moreover, we provide an alternative simplified proof for the uniqueness of weak solutions for r greater than or equal to 3 that holds without imposing any restrictions on viscosity or Forchheimer coefficient. Similar results are also obtained for the case of degenerate mobility and singular potential.