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 $\mathbb{R}^d$ 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.