We consider the incompressible Navier-stokes equations (NS) in R n for n ≥ 2. 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.

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This paper is devoted to the Boussinesq equations that models natural convection in a viscous fluid by coupling Navier-Stokes and heat equations via a zero order approximation. We consider the problem in R n and prove the existence of stationary solutions in critical Besov-Lorentz-Morrey spaces. For that, we prove some estimates for the product of distributions in these spaces, as well as Bernstein inequalities and Mihlin multiplier type results in our setting. Considering in particular the decoupled case, our existence result provides a new class of stationary solutions for the Navier-Stokes equations in critical spaces.

We show bilinear estimates for the Navier-Stokes equations in critical Besov-weak-Morrey (BWM) spaces that contain the so-called Besov-Morrey (BM) spaces. Our estimates employ only the norm of the natural persistence space and do not use auxiliary norms like, e.g., Kato time-weighted norms. As a corollary, we obtain the uniqueness of mild solutions in the class of continuous functions from [0, ∞) to critical BWM-spaces and, in particular, to BM-spaces. For our purposes, we need to show interpolation properties, heat semigroup estimates, and a characterization of preduals (of BWM-spaces) that are Besov-type spaces based on Lorentz-block ones. Another ingredient is a product estimate in our setting.

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