We show that $L^2$ energy estimates combined with Cauchy integral formula for holomorphic functions can provide bounds for higher-order derivatives of smooth solutions of Navier-Stokes equations. We then extend this principle to weak solutions to improve regularization rates obtained by standard energy methods.
"On a method of holomorphic functions to obtain sharp regularization rates of weak solutions of Navier-Stokes equations." Methods Appl. Anal. 14 (4) 345 - 354, December 2007.