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Strong well-posedness and separation properties for a bulk-surface convective Cahn--Hilliard system with singular potentials

Patrik Knopf, Jonas Stange

TL;DR

This work develops a rigorous mathematical theory for a bulk-surface convective Cahn–Hilliard system with singular potentials. By employing a Moreau–Yosida/Yosida regularization of the convex subdifferentials, it first proves the existence of global weak solutions, then establishes uniqueness and continuous dependence on velocity fields and initial data. Under enhanced regularity assumptions, the authors obtain higher regularity and the existence of strong solutions, and they prove strict separation properties for logarithmic potentials in both 2D and 3D. The results extend previous analyses for regular potentials to the singular setting and cover a broad class of dynamic bulk-surface boundary conditions, providing a solid foundation for the mathematical analysis of convex-nonlinearity-driven phase-field models with convective transport and bulk-surface coupling.

Abstract

This paper addresses the well-posedness of a general class of bulk-surface convective Cahn--Hilliard systems with singular potentials. For this model, we first prove the existence of a global-in-time weak solution by approximating the singular potentials via a Yosida regularization, applying the corresponding results for regular potentials, and eventually passing to the limit in this approximation scheme. Then, we prove the uniqueness of weak solutions and their continuous dependence on the velocity fields and the initial data. Afterwards, assuming additional regularity of the domain as well as the velocity fields, we establish higher regularity properties of weak solutions and eventually the existence of strong solutions. In the end, we discuss strict separation properties for logarithmic type potentials in both two and three dimensions.

Strong well-posedness and separation properties for a bulk-surface convective Cahn--Hilliard system with singular potentials

TL;DR

This work develops a rigorous mathematical theory for a bulk-surface convective Cahn–Hilliard system with singular potentials. By employing a Moreau–Yosida/Yosida regularization of the convex subdifferentials, it first proves the existence of global weak solutions, then establishes uniqueness and continuous dependence on velocity fields and initial data. Under enhanced regularity assumptions, the authors obtain higher regularity and the existence of strong solutions, and they prove strict separation properties for logarithmic potentials in both 2D and 3D. The results extend previous analyses for regular potentials to the singular setting and cover a broad class of dynamic bulk-surface boundary conditions, providing a solid foundation for the mathematical analysis of convex-nonlinearity-driven phase-field models with convective transport and bulk-surface coupling.

Abstract

This paper addresses the well-posedness of a general class of bulk-surface convective Cahn--Hilliard systems with singular potentials. For this model, we first prove the existence of a global-in-time weak solution by approximating the singular potentials via a Yosida regularization, applying the corresponding results for regular potentials, and eventually passing to the limit in this approximation scheme. Then, we prove the uniqueness of weak solutions and their continuous dependence on the velocity fields and the initial data. Afterwards, assuming additional regularity of the domain as well as the velocity fields, we establish higher regularity properties of weak solutions and eventually the existence of strong solutions. In the end, we discuss strict separation properties for logarithmic type potentials in both two and three dimensions.
Paper Structure (21 sections, 9 theorems, 41 equations)

This paper contains 21 sections, 9 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. State of the art and related literature.
  3. Goals and novelties of this paper.
  4. Structure of this paper.
  5. Notation, assumptions and preliminaries
  6. Notation
  7. Assumptions
  8. Preliminaries
  9. Main results
  10. Weak solutions for possibly singular potentials
  11. Uniqueness and continuous dependence on the initial data
  12. Strong solutions for singular potentials
  13. Separation properties for logarithmic type potentials
  14. Existence of weak solutions
  15. Maximal monotone graphs and the Yosida approximation
  16. ...and 6 more sections

Key Result

Theorem 3.4

(Existence of weak solutions of EQ:SYSTEM) Suppose that the assumptions ASSUMP:1-ASSUMP:MOBILITY and S1-S2 hold. Let $K,L\in[0,\infty]$, let $\left( \phi_0 , \psi_0 \right)\in\mathcal{H}^1_{K,\alpha}$ be an arbitrary initial datum satisfying cond:init, and let $\boldsymbol{v}\in L^2(0,T;\mathbf{L}^ and the equations are fulfilled in the strong sense. If additionally S4 holds, we even have

Theorems & Definitions (17)

  • Remark 2.1
  • Definition 3.1: Weak solutions of system \ref{['EQ:SYSTEM']}
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • Theorem 3.6
  • Theorem 3.7
  • Remark 3.8
  • Remark 3.9
  • ...and 7 more