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

Manika Bag, Sheetal Dharmatti, Manil T Mohan

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.

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

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.

Paper Structure

This paper contains 10 sections, 10 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Literature Survey
  3. Motivation for the model, difficulties and novelties
  4. Organization of the paper
  5. Mathematical Preliminaries
  6. Linear and nonlinear operators
  7. Weak solutions and energy equality: regular potential and non-degenerate mobility
  8. Energy Equality
  9. Uniqueness
  10. Weak solution and energy equality: degenerate mobility and logarithmic potential

Key Result

Theorem 3.1

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^3$, and the initial condition $(\boldsymbol{u}_0, \varphi_0) \in \mathbb{H}\times\mathrm{H}^1$, $\mathbb{U}\in \mathrm{L}^2(0, T; \mathbb{V}_{\mathrm{div}}')$ and $F$ satisfies the Assumption prop of F. Then there exists a Leray-Hopf weak soluti

Theorems & Definitions (31)

  • Definition 1
  • Theorem 3.1: Leray-Hopf weak solution
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Remark 1
  • proof : Proof of Theorem \ref{['LH weak sol']}
  • Remark 2
  • Lemma 1
  • ...and 21 more