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

TL;DR

This paper develops new bilinear estimates and establishes the uniqueness of mild solutions for Navier-Stokes equations within critical function spaces, expanding the understanding of solution behavior without relying on auxiliary norms.

Contribution

It introduces a general framework for bilinear estimates in critical spaces, including Besov, Morrey, and Besov-Morrey spaces, and demonstrates uniqueness results in Besov-weak-Herz spaces, not previously available.

Findings

Established bilinear estimates in various critical spaces

Proved uniqueness of mild solutions in these spaces

Extended results to Besov-weak-Herz spaces

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.