Global well-posedness of the fractional dissipative system in the framework of variable Fourier--Besov spaces

TL;DR

This paper establishes the global well-posedness of fractional dissipative systems, including Navier--Stokes and Keller--Segel equations, within variable Fourier--Besov spaces, advancing the understanding of fractional PDEs with variable regularity.

Contribution

It introduces the use of variable Fourier--Besov spaces to prove global well-posedness for fractional dissipative PDEs, including Navier--Stokes and Keller--Segel systems.

Findings

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

Established global well-posedness for 3D generalized Navier--Stokes equations.

Demonstrated global well-posedness for the 3D fractional Keller--Segel system.

Abstract

In this paper, we are concerned with the well-posed issues of the fractional dissipative 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 fractional heat equation. Such estimates are fundamental for solving certain dissipative PDE's of fractional type. As an applications, we prove global well-posedness in variable Fourier--Besov spaces for the 3D generalized incompressible Navier--Stokes equations and the 3D fractional Keller--Segel system.