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$H^\infty$-calculus for the Stokes operator with Hodge, Navier, and Robin boundary conditions on unbounded domains

Peer Christian Kunstmann

TL;DR

The paper analyzes the Stokes operator on unbounded uniform $C^{2,1}$-domains under Hodge, Navier, and Robin boundary conditions. It first establishes a strong functional calculus for the Hodge Laplacian, including a bounded Hörmander calculus and bounded $H^$-calculus on solenoidal spaces, together with Gaussian bounds for the semigroup. It then treats Navier and Robin boundary conditions as lower-order perturbations of the Hodge Stokes operator, proving bounded $H^$-calculus and maximal $L^p$-regularity for the Robin-Stokes operators in both $L^q_\sigma( ext{Ω})$ and $ ilde{L}^q_\sigma( ext{Ω})$ spaces, and identifying fractional domain spaces. The results hold under suitable domain invariance assumptions, and extend to certain non-Helmholtz domains, yielding precise domain characterizations and resolvent estimates that underpin stability and regularity in fluid-structure problems with slip-type boundary conditions.

Abstract

We study the Stokes operator with Hodge, Navier, and Robin boundary conditions on domains $Ω\subseteq\mathbb{R}^d$ that are uniformly $C^{2,1}$. Starting with the Hodge Laplacian we etablish a bounded Hörmander functional calculus for the Stokes operator with Hodge boundary conditions. This entails a Hörmander functional calculus and boundedness of the $H^\infty$-calculus in spaces of soleniodal vector fields for the Stokes operator with Hodge boundary conditions. We then establish boundedness of the $H^\infty$-calculus for Stokes operators with Navier type conditions via Robin type perturbations of Hodge boundary conditions. This implies maximal $L^p$-regularity for these operators and results on fractional domain spaces. Our results cover certain non-Helmholtz domains.

$H^\infty$-calculus for the Stokes operator with Hodge, Navier, and Robin boundary conditions on unbounded domains

TL;DR

The paper analyzes the Stokes operator on unbounded uniform -domains under Hodge, Navier, and Robin boundary conditions. It first establishes a strong functional calculus for the Hodge Laplacian, including a bounded Hörmander calculus and bounded -calculus on solenoidal spaces, together with Gaussian bounds for the semigroup. It then treats Navier and Robin boundary conditions as lower-order perturbations of the Hodge Stokes operator, proving bounded -calculus and maximal -regularity for the Robin-Stokes operators in both and spaces, and identifying fractional domain spaces. The results hold under suitable domain invariance assumptions, and extend to certain non-Helmholtz domains, yielding precise domain characterizations and resolvent estimates that underpin stability and regularity in fluid-structure problems with slip-type boundary conditions.

Abstract

We study the Stokes operator with Hodge, Navier, and Robin boundary conditions on domains that are uniformly . Starting with the Hodge Laplacian we etablish a bounded Hörmander functional calculus for the Stokes operator with Hodge boundary conditions. This entails a Hörmander functional calculus and boundedness of the -calculus in spaces of soleniodal vector fields for the Stokes operator with Hodge boundary conditions. We then establish boundedness of the -calculus for Stokes operators with Navier type conditions via Robin type perturbations of Hodge boundary conditions. This implies maximal -regularity for these operators and results on fractional domain spaces. Our results cover certain non-Helmholtz domains.

Paper Structure

This paper contains 20 sections, 34 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Boundary conditions
  4. Regularity of domains
  5. Function spaces and Helmholtz decompositions
  6. Maximal $L^p$-regularity, $H^\infty$-functional calculus, and Hörmander functional calculus
  7. The Hodge Laplacian on uniform $C^{2,1}$-domains
  8. The operator
  9. Gaussian bounds
  10. The Stokes operator with Hodge boundary conditions
  11. Invariance for $q=2$
  12. Invariance for $q\neq 2$
  13. Robin Stokes as perturbations of Hodge Stokes
  14. Estimates for resolvent problems
  15. The Robin Stokes operator in $L^q_\sigma(\Omega)$
  16. ...and 5 more sections

Key Result

Theorem 2.4

Let $\Omega\subseteq\mathbb{R}^d$ be a uniform $C^1$-domain and $1<q<\infty$. Then and the correponding projection $\widetilde{P}_q$ in $\widetilde{L}^q(\Omega)^d$ satisfies $(\widetilde{P}_q)'=\widetilde{P}_{q'}$. Moreover, $C^\infty_{c,\sigma}(\Omega)$ is dense in $\widetilde{L}^q_\sigma(\Omega)$ for the norm of $\widetilde{L}^q(\Omega)^d$ and one has the annihilator relations and in a canonic

Theorems & Definitions (69)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Corollary 3.5
  • ...and 59 more