On Kato's square root property for the generalized Stokes operator

We establish the Kato square root property for the generalized Stokes operator on Rd with bounded measurable coefficients. More precisely, we identify the domain of the square root of Au := − div(μ∇u) + ∇ϕ, div(u) = 0, with the space of divergence-free H1-vector fields and further prove the estimates ∥A1/2u∥L2 ≃ ∥∇u∥L2 . As an application we show that A1/2 depends holomorphically on the coefficients μ. Besides the boundedness and measurability as well as an ellipticity condition on μ, there are no requirements on the coefficients.