$H^{\infty }$-calculus for the Stokes operator with Hodge, Navier, and Robin boundary conditions on unbounded domains

Abstract

We study the Stokes operator with Hodge, Navier, and Robin boundary conditions on domains $\Omega \subseteq {\mathbb{R}}^d$ that are uniformly $C^{2,1}$. Starting with the Hodge Laplacian we etablish a bounded Hörmander functional calculus for the Stokes operator with Hodge boundary conditions. This entails a Hörmander functional calculus and boundedness of the $H^{\infty }$-calculus in spaces of soleniodal vector fields for the Stokes operator with Hodge boundary conditions. We then establish boundedness of the $H^{\infty }$-calculus for Stokes operators with Navier type conditions via Robin type perturbations of Hodge boundary conditions. This implies maximal $L^p$-regularity for these operators and results on fractional domain spaces. Our results cover certain non-Helmholtz domains.