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On $L^p$-semigroup to Stokes equation with dynamic slip boundary condition in the half-space

Dalibor Pražák, Michael Zelina

TL;DR

This work analyzes the evolutionary Stokes system in the half-space with a dynamic slip boundary condition, establishing $L^p$-well-posedness and sharp regularity via an explicit resolvent formula obtained through a tangential Fourier transform. By leveraging $\mathcal{H}^{\infty}$-calculus and Mikhlin multipliers, the authors derive optimal $W^{2,p}$ and $W^{1,p}$ estimates for the stationary problem and prove the existence of an analytic semigroup on $L^p(\Omega)\times L^p(\Gamma)$. The core contribution is a complete $L^p$-theory for weak and strong solutions, including an explicit representation of the resolvent in terms of a fundamental solution $m_0(\lambda,\xi,y)$ and a boundary-correcting term, along with a precise description of the generator’s domain. These results enable sharp regularity for linearized Stokes problems with dynamic boundary conditions and pave the way for bootstrap analyses of nonlinear evolutions in unbounded domains.

Abstract

We consider evolutionary Stokes system, coupled with the so-called dynamic slip boundary condition, in the simple geometry of a $d$-dimensional half-space. Using the Fourier transform, we obtain an explicit formula for the resolvent. Optimal regularity estimates and existence of analytic semigroup in the $L^p$-setting are then deduced using the methods of $\mathcal{H}^{\infty}$-calculus.

On $L^p$-semigroup to Stokes equation with dynamic slip boundary condition in the half-space

TL;DR

This work analyzes the evolutionary Stokes system in the half-space with a dynamic slip boundary condition, establishing -well-posedness and sharp regularity via an explicit resolvent formula obtained through a tangential Fourier transform. By leveraging -calculus and Mikhlin multipliers, the authors derive optimal and estimates for the stationary problem and prove the existence of an analytic semigroup on . The core contribution is a complete -theory for weak and strong solutions, including an explicit representation of the resolvent in terms of a fundamental solution and a boundary-correcting term, along with a precise description of the generator’s domain. These results enable sharp regularity for linearized Stokes problems with dynamic boundary conditions and pave the way for bootstrap analyses of nonlinear evolutions in unbounded domains.

Abstract

We consider evolutionary Stokes system, coupled with the so-called dynamic slip boundary condition, in the simple geometry of a -dimensional half-space. Using the Fourier transform, we obtain an explicit formula for the resolvent. Optimal regularity estimates and existence of analytic semigroup in the -setting are then deduced using the methods of -calculus.
Paper Structure (7 sections, 9 theorems, 101 equations)

This paper contains 7 sections, 9 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Notation and main results
  3. Fundamental solution
  4. Reducing the equations
  5. Fourier transform
  6. Multipliers and $L^p$-bounds
  7. Proof of the main results

Key Result

Theorem 2.1

Let $\alpha \ge0$, $p \in (1, + \infty)$, $\bm f \in L^p_{\sigma,n}(\Omega)$ and $\bm h \in L^p_{n}(\Gamma)$. Then for any complex $\lambda$ with $\mathfrak{Re}\, \lambda > 0$ there exists unique $\bm u \in W^{1,p}_{\sigma,n}(\Omega)$ a weak solution to Res:inside--Res:bound. If moreover $|\lambda| where the constant $C_0$ depends on $d$, $p$ and $\omega$, but is independent of $\lambda$ and $\al

Theorems & Definitions (18)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 8 more