On a bilinear estimate in weak-Morrey spaces and uniqueness for Navier–Stokes equations

Abstract

This paper is concerned with the continuity of the bilinear term B associated with the mild formulation of the Navier–Stokes equations. We provide a new proof for the continuity of B in critical weak-Morrey spaces without using auxiliary norms of Besov type neither Kato time-weighted norms. As a byproduct, we reobtain the uniqueness of mild solutions in the class of continuous functions from [0,T) to critical Morrey spaces. Our proof consists in estimates in block spaces (based on Lorentz spaces) that are preduals of Morrey–Lorentz spaces. For that, we need to obtain properties like interpolation of operators, duality, Hölder and Young type inequalities in such block spaces.