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Diffuse Interface Model for Two-Phase Flows on Evolving Surfaces with Different Densities: Local Well-Posedness

Helmut Abels, Harald Garcke, Andrea Poiatti

TL;DR

The work develops a diffuse-interface model for two-phase flows with non-matched densities on evolving surfaces and proves local well-posedness of strong solutions under prescribed surface evolution. It combines a rational-thermodynamics derivation of a Cahn–Hilliard–Navier–Stokes system on Γ(t) with a surface Piola transform and an evolving-space functional framework to pull back the problem to a fixed reference surface. A fixed-point argument relies on maximal regularity results for the surface Stokes operator with variable viscosity and for the Cahn–Hilliard part, culminating in a short-time existence and uniqueness theorem together with a separation property for strongly separated initial data. The results lay the groundwork for global-in-time analysis and numerical studies of coupled diffuse-interface dynamics on moving manifolds, with potential applications to membrane and interface dynamics in evolving geometries.

Abstract

A Cahn-Hilliard-Navier-Stokes system for two-phase flow on an evolving surface with non-matched densities is derived using methods from rational thermodynamics. For a Cahn-Hilliard energy with a singular (logarithmic) potential short time well-posedness of strong solutions together with a separation property is shown, under the assumption of a priori prescribed surface evolution. The problem is reformulated with the help of a pullback to the initial surface. Then a suitable linearization and a contraction mapping argument for the pullback system are used. In order to deal with the linearized system, it is necessary to show maximal $L^2$-regularity for the surface Stokes operator in the case of variable viscosity and to obtain maximal $L^p$-regularity for the linearized Cahn-Hilliard system.

Diffuse Interface Model for Two-Phase Flows on Evolving Surfaces with Different Densities: Local Well-Posedness

TL;DR

The work develops a diffuse-interface model for two-phase flows with non-matched densities on evolving surfaces and proves local well-posedness of strong solutions under prescribed surface evolution. It combines a rational-thermodynamics derivation of a Cahn–Hilliard–Navier–Stokes system on Γ(t) with a surface Piola transform and an evolving-space functional framework to pull back the problem to a fixed reference surface. A fixed-point argument relies on maximal regularity results for the surface Stokes operator with variable viscosity and for the Cahn–Hilliard part, culminating in a short-time existence and uniqueness theorem together with a separation property for strongly separated initial data. The results lay the groundwork for global-in-time analysis and numerical studies of coupled diffuse-interface dynamics on moving manifolds, with potential applications to membrane and interface dynamics in evolving geometries.

Abstract

A Cahn-Hilliard-Navier-Stokes system for two-phase flow on an evolving surface with non-matched densities is derived using methods from rational thermodynamics. For a Cahn-Hilliard energy with a singular (logarithmic) potential short time well-posedness of strong solutions together with a separation property is shown, under the assumption of a priori prescribed surface evolution. The problem is reformulated with the help of a pullback to the initial surface. Then a suitable linearization and a contraction mapping argument for the pullback system are used. In order to deal with the linearized system, it is necessary to show maximal -regularity for the surface Stokes operator in the case of variable viscosity and to obtain maximal -regularity for the linearized Cahn-Hilliard system.
Paper Structure (15 sections, 8 theorems, 230 equations)

This paper contains 15 sections, 8 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Derivation of the model
  3. Surface Piola transform and functional setting
  4. Flow map and Surface Piola Transform
  5. Evolving Banach and Hilbert spaces
  6. Assumptions and main result
  7. Equivalent formulation for a divergence-free velocity
  8. Pullback equations
  9. Short time existence of a regular solution to the pullback problem
  10. Proof of Theorem \ref{['thm1']}
  11. Proof of Proposition \ref{['prop1']}
  12. Proof of Proposition \ref{['prop2']}
  13. Proof of Theorem \ref{['strong1a']}
  14. On the regularity for a Laplace equation on evolving surfaces
  15. Computations for the pullback equations

Key Result

Theorem 4.1

Assume the regularity assumptions on $\Phi_t^n$ stated in Section regflowmap1, and assumptions l1-ro for $\Psi$, $\nu$, and $\rho$. Let $\mathbf{v}_0 \in \mathbf{H}^1(\Gamma_0)$ and $\varphi_0\in B_{p,q}^{4-\frac{4}{q}}(\Gamma_0)$, for some $2<q\leq 4$ and $p>4$. Then there exists a $\widetilde{T}\i and it satisfies mainp in the almost everywhere sense. Furthermore, if $\left\Vert\varphi_0\right\V

Theorems & Definitions (17)

  • Theorem 4.1: Short time well-posedness of strong solutions
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Theorem 7.1
  • Lemma 7.2
  • Remark 7.3
  • proof
  • Lemma 7.4
  • Remark 7.5
  • ...and 7 more