# On critical spaces for the Navier-Stokes equations

## TL;DR

This paper applies the abstract theory of critical spaces to the Navier-Stokes equations in bounded domains, unifying and extending existing $L_p$-$L_q$ results, and characterizes the associated Stokes operators.

## Contribution

It demonstrates that the strong and weak Stokes operators with Navier boundary conditions admit an $ ext{-}$-calculus and identifies their interpolation spaces, extending the theoretical framework.

## Key findings

- Stokes operators with Navier conditions admit an $ ext{-}$-calculus.
- Interpolation spaces of these operators are explicitly identified.
- Unified approach simplifies existing $L_p$-$L_q$ analysis.

## Abstract

The abstract theory of critical spaces developed in [22] and [20] is applied to the Navier-Stokes equations in bounded domains with Navier boundary conditions as well as no-slip conditions. Our approach unifies, simplifies and extends existing work in the $L_p$-$L_q$ setting, considerably. As an essential step, it is shown that the strong and weak Stokes operators with Navier conditions admit an $\mathcal{H}^\infty$-calculus with $\mathcal{H}^\infty$-angle 0, and the real and complex interpolation spaces of these operators are identified.