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Global well-posedness of strong solutions to a bulk-surface Navier-Stokes-Cahn-Hilliard model with non-degenerate mobilities in two dimensions

Jonas Stange

TL;DR

The paper proves the global well-posedness of strong solutions for a two-dimensional bulk–surface Navier–Stokes–Cahn–Hilliard model with non-degenerate mobilities and dynamic bulk–surface coupling. It develops new regularity theory for convective bulk–surface Cahn–Hilliard equations and a bulk–surface Stokes system with variable coefficients, then constructs global strong solutions via a semi-Galerkin approach combined with Schauder fixed-point arguments and rigorous a priori estimates. Uniqueness and continuous dependence on initial data are shown for $L\in(0,\infty]$, while the case $L=0$ is excluded except in a special compatible setting when $K=L=0$. The results advance the mathematical understanding of coupled bulk-surface diffuse-interface dynamics and provide a rigorous foundation for models of membrane-fluid interactions with mass exchange.

Abstract

We examine a thermodynamically consistent diffuse interface model for bulk-surface viscous fluid mixtures. This model consists of a Navier--Stokes--Cahn--Hilliard model in the bulk coupled to a surface Navier--Stokes--Cahn--Hilliard system on the boundary. In this paper, we address the global well-posedness of strong solutions in the two-dimensional setting, also covering the physically meaningful case of non-degenerate mobility functions. Lastly, we prove the uniqueness of the corresponding strong solutions and their continuous dependence on the initial data. Our approach hinges upon new well-posedness and regularity theory for a convective bulk-surface Cahn--Hilliard equation with non-degenerate mobilities, as well as a bulk-surface Stokes equation with non-constant coefficients.

Global well-posedness of strong solutions to a bulk-surface Navier-Stokes-Cahn-Hilliard model with non-degenerate mobilities in two dimensions

TL;DR

The paper proves the global well-posedness of strong solutions for a two-dimensional bulk–surface Navier–Stokes–Cahn–Hilliard model with non-degenerate mobilities and dynamic bulk–surface coupling. It develops new regularity theory for convective bulk–surface Cahn–Hilliard equations and a bulk–surface Stokes system with variable coefficients, then constructs global strong solutions via a semi-Galerkin approach combined with Schauder fixed-point arguments and rigorous a priori estimates. Uniqueness and continuous dependence on initial data are shown for , while the case is excluded except in a special compatible setting when . The results advance the mathematical understanding of coupled bulk-surface diffuse-interface dynamics and provide a rigorous foundation for models of membrane-fluid interactions with mass exchange.

Abstract

We examine a thermodynamically consistent diffuse interface model for bulk-surface viscous fluid mixtures. This model consists of a Navier--Stokes--Cahn--Hilliard model in the bulk coupled to a surface Navier--Stokes--Cahn--Hilliard system on the boundary. In this paper, we address the global well-posedness of strong solutions in the two-dimensional setting, also covering the physically meaningful case of non-degenerate mobility functions. Lastly, we prove the uniqueness of the corresponding strong solutions and their continuous dependence on the initial data. Our approach hinges upon new well-posedness and regularity theory for a convective bulk-surface Cahn--Hilliard equation with non-degenerate mobilities, as well as a bulk-surface Stokes equation with non-constant coefficients.

Paper Structure

This paper contains 18 sections, 15 theorems, 333 equations.

Table of Contents

  1. Introduction
  2. Mathematical setting
  3. General notation and function spaces
  4. Differential operators in the bulk and on the surface
  5. Bulk-surface product spaces
  6. Spaces of tangential and divergence-free vector fields
  7. Important tools.
  8. Elliptic problems with bulk-surface interaction.
  9. Assumptions
  10. Main results
  11. Existence of a global-in-time strong solution
  12. Uniqueness
  13. The convective bulk-surface Cahn--Hilliard equation with non-degenerate mobility
  14. On the bulk-surface Stokes equation
  15. Non-homogeneous Stokes equation with non-constant viscosity
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let $K\in[0,\infty)$ and $\alpha,\beta\in\mathbb R$ with $\alpha\beta| \Omega | + | \Gamma | \neq 0$. Then there exists a constant $C_P > 0$ depending only on $K,\alpha,\beta$ and $\Omega$ such that for all pairs $(\phi,\psi)\in\mathcal{H}^1_{K,\alpha}$ satisfying $\textnormal{mean} ( \phi , \psi ) = 0$.

Theorems & Definitions (28)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • proof : Proof of Theorem \ref{['Theorem:GlobalExistence']}.
  • ...and 18 more