# On Critical Spaces for the Navier–Stokes Equations

On Critical Spaces for the Navier–Stokes Equations The abstract theory of critical spaces developed in Prüss and Wilke (J Evol Equ, 2017. doi: 10.1007/s00028-017-0382-6 ), Prüss et al. (Critical spaces for quasilinear parabolic evolution equations and applications, 2017) 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 p – $$L_q$$ L q setting, considerably. As an essential step, it is shown that the strong and weak Stokes operators with Navier conditions admit an $$\mathcal {H}^\infty$$ H ∞ -calculus with $$\mathcal {H}^\infty$$ H ∞ -angle 0, and the real and complex interpolation spaces of these operators are identified. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Mathematical Fluid Mechanics Springer Journals

# On Critical Spaces for the Navier–Stokes Equations

, Volume 20 (2) – Oct 5, 2017
23 pages

/lp/springer_journal/on-critical-spaces-for-the-navier-stokes-equations-h0V0VGiMur
Publisher
Springer Journals
Subject
Physics; Fluid- and Aerodynamics; Mathematical Methods in Physics; Classical and Continuum Physics
ISSN
1422-6928
eISSN
1422-6952
D.O.I.
10.1007/s00021-017-0342-5
Publisher site
See Article on Publisher Site

### Abstract

The abstract theory of critical spaces developed in Prüss and Wilke (J Evol Equ, 2017. doi: 10.1007/s00028-017-0382-6 ), Prüss et al. (Critical spaces for quasilinear parabolic evolution equations and applications, 2017) 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 p – $$L_q$$ L q setting, considerably. As an essential step, it is shown that the strong and weak Stokes operators with Navier conditions admit an $$\mathcal {H}^\infty$$ H ∞ -calculus with $$\mathcal {H}^\infty$$ H ∞ -angle 0, and the real and complex interpolation spaces of these operators are identified.

### Journal

Journal of Mathematical Fluid MechanicsSpringer Journals

Published: Oct 5, 2017

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