Bilinear estimates and uniqueness for Navier-Stokes equations in critical Besov-type spaces

Abstract

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.