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

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.