Global well-posedness of the Navier--Stokes equations and the Keller--Segel system in variable Fourier--Besov spaces

TL;DR

This paper establishes the global well-posedness of the 3D Navier--Stokes and Keller--Segel systems within variable Fourier--Besov spaces, advancing the understanding of these PDEs in complex functional frameworks.

Contribution

It introduces the use of variable Fourier--Besov spaces to prove global solutions for Navier--Stokes and Keller--Segel equations, a novel approach in this context.

Findings

Proved linear estimates for heat equations in variable Fourier--Besov spaces.

Established global well-posedness results for Navier--Stokes in these spaces.

Demonstrated similar results for the Keller--Segel system.

Abstract

In this paper, we study the Cauchy problem of the classical incompressible Navier--Stokes equations and the parabolic-elliptic Keller--Segel system in the framework of the Fourier--Besov spaces with variable regularity and integrability indices. By fully using some basic properties of these variable function spaces, we establish the linear estimates in variable Fourier--Besov spaces for the heat equation. Such estimates are fundamental for solving certain PDE's of parabolic type. As an applications, we prove global well-posedness in variable Fourier--Besov spaces for the 3D classical incompressible Navier--Stokes equations and the 3D parabolic-elliptic Keller--Segel system.