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On Kato's Square Root Property for the Generalized Stokes Operator

Luca Haardt, Patrick Tolksdorf

Abstract

We establish the Kato square root property for the generalized Stokes operator on $\mathbb{R}^d$ with bounded measurable coefficients. More precisely, we identify the domain of the square root of $Au := - \operatorname{div}(μ\nabla u) + \nabla φ$, $\operatorname{div}(u) = 0$, with the space of divergence-free $\mathrm{H}^1$-vector fields and further prove the estimate $\|A^{1/2} u \|_{\mathrm{L}^2} \simeq \| \nabla u \|_{\mathrm{L}^2}$. As an application we show that $A^{1/2}$ depends holomorphically on the coefficients $μ$. Besides the boundedness and measurablility as well as an ellipticity condition on $μ$, there are no requirements on the coefficients.

On Kato's Square Root Property for the Generalized Stokes Operator

Abstract

We establish the Kato square root property for the generalized Stokes operator on with bounded measurable coefficients. More precisely, we identify the domain of the square root of , , with the space of divergence-free -vector fields and further prove the estimate . As an application we show that depends holomorphically on the coefficients . Besides the boundedness and measurablility as well as an ellipticity condition on , there are no requirements on the coefficients.

Paper Structure

This paper contains 6 sections, 22 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. $\mathrm{L}^2$-theory of the generalized Stokes resolvent
  3. Reduction to a square function estimate
  4. Principle part approximation
  5. The Carleson measure argument
  6. Holomorphic dependency

Key Result

Theorem 1.2

Let $\mu$ satisfy Assumption Ass: Coefficients. Then $A$ has the square root property, i.e., we have that $\mathop{\mathrm{\mathcal{D}}}\nolimits(A^{1 / 2}) = \mathrm{H}^1_{\sigma} (\mathbb{R}^d)$ and where $C > 0$ only depends on $d$, $\mu_{\bullet}$ and $\mu^{\bullet}$.

Theorems & Definitions (46)

  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • ...and 36 more