On regularity of solutions to the Navier--Stokes equation with initial data in ${\mathrm{B}\mathrm{M}\mathrm{O}}^{-1}$

We prove that any mild solution in the Koch–Tataru space to the incompressible Navier–Stokes equation with initial data in ${\mathrm{B}\mathrm{M}\mathrm{O}}^{-1}$ is weak*-continuous in time, valued in ${\mathrm{B}\mathrm{M}\mathrm{O}}^{-1}$. We also show that the global mild solution vanishes in ${\mathrm{B}\mathrm{M}\mathrm{O}}^{-1}$ at infinity in time.