Space-time derivative estimates of the Koch-Tataru solutions to the nematic liquid crystal system in Besov spaces

In recent paper [7], Y. Du and K. Wang (2013) proved that the global-in-time Koch-Tataru type solution (u,d) to the n-dimensional incompressible nematic liquid crystal flow with small initial data (u0,d0) in BMO−1×BMO has arbitrary space-time derivative estimates in the so-called Koch-Tataru space norms. The purpose of this paper is to show that the Koch-Tataru type solution satisfies the decay estimates for any space-time derivative involving some borderline Besov space norms. More precisely, for the global-in-time Koch-Tataru type solution (u,d) to the nematic liquid crystal flow with initial data (u0,d0)∈BMO−1×BMO and ‖u0‖BMO−1+[d0]BMO≤ε for some small enough ε>0, and for any positive integers k and m, one has‖tk+m2(∂tk∇mu,∂tk∇m∇d)‖L˜∞(R+,B˙∞,∞−1)∩L˜1(R+;B˙∞,∞1)≤ε. Furthermore, we shall give that the solution admits a unique trajectory which is Hölder continuous with respect to space variables.