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Global existence of weak solutions to incompressible anisotropic Cahn-Hilliard-Navier-Stokes system

Azeddine Zaidni, Saad Benjelloun, Radouan Boukharfane

Abstract

We study the anisotropic, incompressible Cahn-Hilliard-Navier-Stokes system with variable density in a bounded smooth domain $Ω\subset \mathbb{R}^d$. This work extends previous results on the isotropic case by incorporating anisotropic surface energy, represented by $\mathfrak{F}= \int_Ω \fracε{2}\, Γ^2(\nabla φ) $. The thermodynamic consistency of this system, as well as its modeling background and physical motivation, has been established in \cite{anderson2000phase,taylor-cahn98, zaidni2024}. Using a Galerkin approximation scheme, we prove the existence of global weak solutions in both two- and three-dimensions $(d=2,3)$. A key ingredient in extending the local existence of approximate solutions to a global one is the application of Bihari's inequality combined with a fixed-point argument.

Global existence of weak solutions to incompressible anisotropic Cahn-Hilliard-Navier-Stokes system

Abstract

We study the anisotropic, incompressible Cahn-Hilliard-Navier-Stokes system with variable density in a bounded smooth domain . This work extends previous results on the isotropic case by incorporating anisotropic surface energy, represented by . The thermodynamic consistency of this system, as well as its modeling background and physical motivation, has been established in \cite{anderson2000phase,taylor-cahn98, zaidni2024}. Using a Galerkin approximation scheme, we prove the existence of global weak solutions in both two- and three-dimensions . A key ingredient in extending the local existence of approximate solutions to a global one is the application of Bihari's inequality combined with a fixed-point argument.

Paper Structure

This paper contains 7 sections, 5 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Energy balance
  4. Regularized logarithmic potential
  5. Global existence of an approximate solution
  6. Proof of theorem \ref{['th1']}
  7. Bihari's Inequality

Key Result

Theorem 2.1

Let $T$ be a positive time. Assume that $\rho_0 \in L^\infty(\Omega)$ with $0 < \rho_* \leq \rho_0 \leq \rho^* < \infty$, $\mathbf{u}_0 \in L^2(\Omega)$, whose divergence vanishes in the weak sense, and $\phi_0 \in H^1(\Omega) \cap L^{\infty}(\Omega)$ such that $\|\phi_0\|_{L^{\infty}(\Omega)} \leq

Theorems & Definitions (8)

  • Theorem 2.1
  • Proposition 5.1
  • proof
  • Proposition 5.2
  • proof
  • Proposition 6.1
  • proof
  • Lemma A.1