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On Cahn-Hilliard-Navier-Stokes equations with Nonhomogeneous Boundary

Manika Bag, Tania Biswas, Sheetal Dharmatti

TL;DR

The paper analyzes the Cahn–Hilliard–Navier–Stokes system with a nonhomogeneous velocity boundary condition $\mathbf{u}=\mathbf{h}$ on a 2D bounded domain, addressing non-autonomous challenges via an elliptic lifting that transfers the problem to homogeneous boundaries. It proves the existence of global weak solutions, continuous dependence and uniqueness, and existence of strong solutions, supported by energy estimates and fixed-point arguments. The approach yields a robust well-posedness theory for CHNS with nonzero boundary data and demonstrates long-time convergence of weak solutions to stationary states in two dimensions. These results provide a rigorous foundation for boundary-control-type studies and enhance understanding of the model's long-time behavior in diffuse-interface fluid mixtures.

Abstract

The evolution of two isothermal, incompressible, immiscible fluids in a bounded domain is governed by Cahn-Hilliard-Navier-Stokes equations (CHNS System). In this work, we study the well-posedness results for the CHNS system with nonhomogeneous boundary condition for the velocity equation. We obtain the existence of global weak solutions in the two-dimensional bounded domain. We further prove the continuous dependence of the solution on initial conditions and boundary data that will provide the uniqueness of the weak solution. The existence of strong solutions is also established in this work. Furthermore, we show that in the two-dimensional case, each global weak solution converges to a stationary solution.

On Cahn-Hilliard-Navier-Stokes equations with Nonhomogeneous Boundary

TL;DR

The paper analyzes the Cahn–Hilliard–Navier–Stokes system with a nonhomogeneous velocity boundary condition on a 2D bounded domain, addressing non-autonomous challenges via an elliptic lifting that transfers the problem to homogeneous boundaries. It proves the existence of global weak solutions, continuous dependence and uniqueness, and existence of strong solutions, supported by energy estimates and fixed-point arguments. The approach yields a robust well-posedness theory for CHNS with nonzero boundary data and demonstrates long-time convergence of weak solutions to stationary states in two dimensions. These results provide a rigorous foundation for boundary-control-type studies and enhance understanding of the model's long-time behavior in diffuse-interface fluid mixtures.

Abstract

The evolution of two isothermal, incompressible, immiscible fluids in a bounded domain is governed by Cahn-Hilliard-Navier-Stokes equations (CHNS System). In this work, we study the well-posedness results for the CHNS system with nonhomogeneous boundary condition for the velocity equation. We obtain the existence of global weak solutions in the two-dimensional bounded domain. We further prove the continuous dependence of the solution on initial conditions and boundary data that will provide the uniqueness of the weak solution. The existence of strong solutions is also established in this work. Furthermore, we show that in the two-dimensional case, each global weak solution converges to a stationary solution.
Paper Structure (8 sections, 9 theorems, 109 equations)

This paper contains 8 sections, 9 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Functional Setup
  4. Linear and Nonlinear Operators
  5. Elliptic Lifting Operator
  6. Existence of a weak solution
  7. Continuous dependence
  8. Strong Solution

Key Result

Proposition 1

Let $\Omega$ be a bounded domain with a smooth boundary in $\mathbb{R}^2$. Assume that $f, g \in \mathrm{H}^1$. Then, there exists a positive constant $C$ such that

Theorems & Definitions (23)

  • Proposition 1
  • Lemma 1
  • Remark 1
  • Theorem 2.1
  • proof
  • Definition 1
  • Remark 2
  • Lemma 2
  • proof
  • Theorem 3.1
  • ...and 13 more