Properties of Besov-Lorentz spaces and application to Navier-Stokes equations

TL;DR

This paper explores properties of Besov-Lorentz spaces and applies them to analyze the behavior of solutions to Navier-Stokes equations, particularly focusing on controlling the set of large value points and establishing global well-posedness.

Contribution

It introduces new analytical tools involving Besov-Lorentz spaces and wavelet techniques to better understand the large value set behavior in Navier-Stokes solutions, leading to a global well-posedness result.

Findings

Properties of Besov-Lorentz spaces are established.

Wavelet and maximum norm methods describe decay and large value sets.

A global well-posedness result for Navier-Stokes equations is obtained.

Abstract

Inspired by Caffarelli-Kohn-Nirenberg, Fefferman and Lin, we try to investigate how to control the set of large value points for the strong solution of Navier-Stokes equations. Besov-Lorentz spaces have multiple indices which can reflect complex changes of the set of the large value points. Hence we consider some properties of Gauss flow, paraproduct flow and couple flow related to the Besov-Lorentz spaces. When dealing with Lorentz index, we use wavelets and maximum norm to describe the decay situation in the binary time ring and to define time-frequency microlocal maximum norm space. We use maximum operator, $\alpha$-triangle inequality and H\"older inequality etc to get accurate estimates. As an application, we get a global wellposedness result of the Navier-Stokes equations where the solution can reflect how the size of the set of large value points changes.