On critical spaces for the Navier-Stokes equations

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.

arXiv comment21 pages

References in corpus (2)

• On convergence of solutions to equilibria for quasilinear parabolic problems
• Critical spaces for quasilinear parabolic evolution equations and applications

Cited by in corpus (6)

• Critical spaces for quasilinear parabolic evolution equations and applications
• $H^{\infty }$-calculus for the surface Stokes operator and applications
• A discrete framework for the interpolation of Banach spaces
• Nonlinear parabolic stochastic evolution equations in critical spaces Part II. Blow-up criteria and instantaneous regularization
• Stochastic Navier-Stokes equations for turbulent flows in critical spaces
• On the Navier-Stokes equations on surfaces