  We consider the incompressible Navier-stokes equations (NS) in
$\mathbb{R}^{n}$ for $n\geq2$. 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, it is proved 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.
