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 $Au:=-\mathrm{div}(\hat{I}¼\nabla u)+\nabla \ddot{I}•$, $\mathrm{div}(u)=0$, with the space of divergence-free ${\mathrm{H}}^1$-vector fields and further prove the estimate $\Vert A^{1/2}u{\Vert }_{{\mathrm{L}}^2}\simeq \Vert \nabla u{\Vert }_{{\mathrm{L}}^2}$. As an application we show that $A^{1/2}$ depends holomorphically on the coefficients $\hat{I}¼$. Besides the boundedness and measurablility as well as an ellipticity condition on $\hat{I}¼$, there are no requirements on the coefficients.