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

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 $Au:=-\mathrm{div}(\mu \mathrm{\nabla }u)+\mathrm{\nabla }\phi$, $\mathrm{div}(u)=0$, with the space of divergence-free ${\mathrm{H}}^1$-vector fields and further prove the estimate $\mathrm{\parallel }{A}^{1\mathrm{/}2}u{\mathrm{\parallel }}_{{\mathrm{L}}^2}\simeq \mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{{\mathrm{L}}^2}$. As an application we show that $A^{1\mathrm{/}2}$ 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.