On critical spaces for the Navier-Stokes equations

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 ${\mathcal{H}}^{\mathrm{\infty }}$-calculus with ${\mathcal{H}}^{\mathrm{\infty }}$-angle 0, and the real and complex interpolation spaces of these operators are identified.