J. Math. Fluid Mech. 20 (2018), 733–755
2017 Springer International Publishing AG
Journal of Mathematical
On Critical Spaces for the Navier–Stokes Equations
Jan Pr¨uss and Mathias Wilke
Abstract. The abstract theory of critical spaces developed in Pr¨uss and Wilke (J Evol Equ, 2017.doi:10.1007/
s00028-017-0382-6), Pr¨uss 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
setting, considerably. As an essential step, it is
shown that the strong and weak Stokes operators with Navier conditions admit an H
-calculus with H
-angle 0, and the
real and complex interpolation spaces of these operators are identiﬁed.
Keywords. Navier-Stokes equations, Navier boundary conditions, perfect-slip boundary conditions, critical spaces, functional
calculus, weak and strong Stokes operator.
There is no clear dseﬁnition of ‘critical spaces’ for PDEs in the literature. One possibility would be ‘the
largest class of initial data such that the given PDE is uniquely solvable or well-posed in a prescribed
class of functions’. This ‘deﬁnition’ has the disadvantage that by only changing the sign of one term of
a PDE, the ‘critical space’ may change dramatically; so it is by no means a robust deﬁnition. In the
literature, critical spaces are often introduced as the scaling invariant spaces, if the underlying PDE has
such a scaling. Apparently, this seems to require that each of such equation has to be studied separately.
If there is no scaling, it is not clear what to do.
In our innovative approach, we start with a given functional analytic setting, the ‘class of functions’
and ﬁnd a space—we call it the critical space—such that the problem is well-posed for initital values in
this space. By means of counterexamples we can show that this is generically the largest such class. Also,
we can prove that this space is to some extent independent of the setting, more precisely, independent of
the natural scale of function spaces involved. Thirdly, we can also show that the critical spaces are scaling
invariant, if the original PDE admits a scaling, see Pr¨uss et al.  for these general facts. Our methods
apply to a variety of problems, which besides the Navier–Stokes equations include Keller–Segel models in
chemotaxis, Leslie–Ericksen equations for liquid crystals, Nernst–Planck–Poisson systems in electrochem-
istry, reaction-convection-diﬀusion systems, MHD equations, and quasi-geostrophic equations. We refer
to our forthcoming paper Pr¨uss et al. , as well as to Pr¨uss  for the quasi-geostrophic equations.
In this paper we apply this abstract approach to boundary value problems for the Navier–Stokes
u − Δu + u ·∇u + ∇π =0,t>0,x∈ Ω,
div u =0,t>0,x∈ Ω,
u(0) = u
in a bounded domain Ω ⊂ R
with boundary Σ := ∂Ω ∈ C
, where u is the velocity ﬁeld and π means
the pressure. We are mainly concerned with Navier boundary conditions
u · ν =0,P
((∇u + ∇u
)ν)+αu =0 on Σ