On the regularizing rate estimates of Koch–Tataru's solution to the Navier–Stokes equations

Abstract

Koch and Tataru (Adv. Math. 157 (2001), 22–35) showed that the Cauchy problem of the Navier–Stokes equations has a time-local mild solution, when the initial velocity a∈vmo⁻¹ (or, a∈bmo⁻¹ and ‖a‖_bmo⁻¹ is small enough). The purpose of this paper is to estimate the regularizing rates for the higher-order derivatives of the mild solution. As an application of these estimates, it is proved that the solution is analytic in space variables. Moreover, it is also shown that the Serrin's condition leads to the spatial analyticity of the solution.