On bilinear estimates and critical uniqueness classes for Navier-Stokes equations

Abstract

We are concerned with bilinear estimates and uniqueness of mild solutions for the Navier-Stokes equations in critical spaces. For that, we construct general settings in which estimates for the bilinear term of the mild formulation hold true without using auxiliary norms such as Kato time-weighted ones. We first obtain necessary conditions in abstract critical spaces and then consider further structures to obtain the estimates in general classes of Besov, Morrey and Besov-Morrey spaces based on Banach spaces. Examples of applications are provided in different spaces as well as for other PDEs. In particular, as far as we know, the bilinear estimate and the uniqueness property obtained in the framework of Besov-weak-Herz spaces are not available in the existing literature. The proofs are mainly based on characterizations and estimates on the corresponding predual spaces.