We consider the incompressible Navier-Stokes equations (NS) in
for
.
Global well-posedness is proved in critical Besov-weak-Herz spaces (BWH-spaces)
that consist in Besov spaces based on weak-Herz spaces. These spaces are
larger than some critical spaces considered in previous works for NS. For our
purposes, we need to develop a basic theory for BWH-spaces containing
properties and estimates such as heat semigroup estimates, embedding theorems,
interpolation properties, among others. In particular, we prove a characterization of
Besov-weak-Herz spaces as interpolation of Sobolev-weak-Herz ones, which is key in
our arguments. Self-similarity and asymptotic behavior of solutions are also
discussed. Our class of spaces and its properties developed here could also be
employed to study other PDEs of elliptic, parabolic and conservation-law
type.
