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

TL;DR

This paper proves the Kato square root property for a generalized Stokes operator with measurable coefficients, establishing domain identification and holomorphic dependence on coefficients, without regularity assumptions.

Contribution

It extends the Kato square root property to the generalized Stokes operator with minimal coefficient regularity, including domain characterization and holomorphic dependence.

Findings

Domain of the square root identified with divergence-free H^1 fields

Establishment of the estimate A^{1/2} u  abla u in L^2

Holomorphic dependence of A^{1/2} on coefficients  

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 $Au := - \operatorname{div}(\mu \nabla u) + \nabla \phi$, $\operatorname{div}(u) = 0$, 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 $\mu$. Besides the boundedness and measurablility as well as an ellipticity condition on $\mu$, there are no requirements on the coefficients.