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

作者：Liu Qiao^*

来源：Journal of Differential Equations, 2015, 258(12): 4368-4397.

DOI：10.1016/j.jdo.2015.01.036

• Nematic liquid crystal flow
• Navier-Stokes equations
• Regularity
• Littlewood-Paley decomposition
• Besov space
• Trajectory

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 (u(0), do) in BMO-1 x 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 (u(0), d(0)) is an element of BMO-1 x BMO and parallel to u(0)parallel to(BMO-1) + [d(0)](BMO) <= epsilon. a for some small enough epsilon > 0, and for any positive integers k and m, one has <br xmlns:set="http://exslt.org/sets">parallel to tk+m/2(partial derivative tk del mu, partial derivative tk del m del d parallel to (L) over tilde (infinity) (R+, (B)over dot(infinity,infinity)(-1)) boolean AND (R+; (B)over dot(infinity infinity)(1)) <= epsilon
Furthermore, we shall give that the solution admits a unique trajectory which is Holder continuous with respect to space variables.

• 出版日期2015-6-15
• 单位湖南师范大学