Space–time regularity of the Koch & Tataru solutions to Navier–Stokes equations

In Koch & Tataru (2001), the authors have proved local and global well-posedness of the Navier–Stokes equations with small initial data u0∈BMO−1u0∈BMO−1. And then the spatial analyticity of the Koch & Tataru solution stated as tk/2∇ku∈XT∗tk/2∇ku∈XT∗ for any positive integer kk has been presented by Germain–Pavlovic’–Staffilani (2007), where XT∗XT∗ is the space defined by Koch & Tataru (2001), (see also Definition 1.2). In this paper, we shall present the space–time regularity. More precisely, for any positive integer m,k, we have tm+k2∂tm∇ku∈XT∗.