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

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.