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

2024/10/24 by Luca Haardt, Haardt, Luca, Patrick Tolksdorf +1
Computer Science · Mathematics · Engineering · #Advanced Mathematical Modeling in Engineering #Navier-Stokes equation solutions #Advanced Numerical Methods in Computational Mathematics

paper · pdf · doi:10.48550/arxiv.2410.18787

Abstract

We establish the Kato square root property for the generalized Stokes operator on ℝ^d 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 estimate ‖A^1/2 u ‖_L² ≃ ‖ ∇ u ‖_L². As an application we show that A^1/2 depends holomorphically on the coefficients μ. Besides the boundedness and measurablility as well as an ellipticity condition on μ, there are no requirements on the coefficients.

Related

• Kato square root problem with unbounded leading coefficients
• The Kato square root problem on an arbitrary domain of ℝ^d
• Kato's square root problem in Banach spaces
• The Kato square root problem for weighted parabolic operators
• A non-local approach to the generalized Stokes operator with bounded\n measurable coefficients