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Maximal regularity of Stokes problem with dynamic boundary condition -- Hilbert setting

Tomáš Bárta, Paige Davis, Petr Kaplický

TL;DR

This work studies maximal $L^q$-regularity for the linear Stokes problem with a dynamic boundary condition by formulating it as an abstract Cauchy problem on a Hilbert product space and proving that the associated operator $\mathcal{A}$ is sectorial and generates an analytic semigroup. Using elliptic regularity and a Leray projection framework, the authors prove density, closedness, and resolvent bounds, which yield analytic semigroup generation and, via maximal $L^q$-regularity, strong time-space regularity for the velocity and pressure. The results establish existence, uniqueness, and optimal regularity of weak solutions for all $q\in(1,\infty)$ with precise a priori estimates, provided suitable initial data in the real interpolation spaces. This provides a rigorous foundation for the maximal regularity theory of Stokes problems with dynamic boundary conditions in the Hilbert setting, enabling analysis of nonlinear extensions and pressure reconstruction. The approach synthesizes elliptic regularity, operator theory, and semigroup methods to address the coupling between interior and boundary dynamics.

Abstract

For the evolutionary Stokes problem with dynamic boundary conditions, we show the maximal regularity of weak solutions in time. Due to the characterization of $R$-sectorial operators on Hilbert spaces, the proof reduces to identifying the appropriate functional analytic setting and proving that the corresponding operator is sectorial, i.e., that it generates an analytic semigroup.

Maximal regularity of Stokes problem with dynamic boundary condition -- Hilbert setting

TL;DR

This work studies maximal -regularity for the linear Stokes problem with a dynamic boundary condition by formulating it as an abstract Cauchy problem on a Hilbert product space and proving that the associated operator is sectorial and generates an analytic semigroup. Using elliptic regularity and a Leray projection framework, the authors prove density, closedness, and resolvent bounds, which yield analytic semigroup generation and, via maximal -regularity, strong time-space regularity for the velocity and pressure. The results establish existence, uniqueness, and optimal regularity of weak solutions for all with precise a priori estimates, provided suitable initial data in the real interpolation spaces. This provides a rigorous foundation for the maximal regularity theory of Stokes problems with dynamic boundary conditions in the Hilbert setting, enabling analysis of nonlinear extensions and pressure reconstruction. The approach synthesizes elliptic regularity, operator theory, and semigroup methods to address the coupling between interior and boundary dynamics.

Abstract

For the evolutionary Stokes problem with dynamic boundary conditions, we show the maximal regularity of weak solutions in time. Due to the characterization of -sectorial operators on Hilbert spaces, the proof reduces to identifying the appropriate functional analytic setting and proving that the corresponding operator is sectorial, i.e., that it generates an analytic semigroup.
Paper Structure (9 sections, 13 theorems, 66 equations)

This paper contains 9 sections, 13 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Notation and functional analytic setting
  3. Notation and function spaces
  4. Definition of the operator $\mathcal{A}$
  5. Regularity theory for the elliptic problem
  6. Proof of the main theorem
  7. Uniqueness of the weak solutions
  8. The operator $\mathcal{A}$ generates an analytic semigroup
  9. Proof of Theorem \ref{['thm:main']}

Key Result

Lemma 2

Let $\Omega$ be nonaxisymmetric. There exists $\alpha_0<0$ such that for all $u\in H^1(\Omega)$ with $u\cdot\nu=0$ on $\partial\Omega$

Theorems & Definitions (31)

  • Definition 1
  • Lemma 2
  • Theorem 3
  • Lemma 4
  • Definition 5
  • Definition 6
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 21 more