Space–time regularity of the mild solutions to the incompressible generalized Navier–Stokes equations with small rough initial data

In recent paper of Li and Zhai (2010), the authors proved the global-in-time existence and spatial regularity of mild   solutions to the nn-dimensional incompressible generalized Navier–Stokes equations with small initial data u0u0 in Qα;∞β,−1(Rn)≔∇⋅(Qαβ(Rn))n, β∈(12,1] and α∈[0,β)α∈[0,β). In this paper, by using the Fourier localization method, we shall show that the mild solution presented by Li and Zhai (2010) satisfies the decay estimates for any space–time derivative involving some borderline Besov space norms. Moreover, the solution has a unique trajectory which is Hölder continuous with respect to space variables.