Existence and analyticity of mild solutions for the 3D rotating Navier–Stokes equations

Abstract

We study the three-dimensional Navier–Stokes equations in the rotational framework. By using the Littlewood–Paley analysis technique and the dispersive estimates for the Coriolis linear group \(\{e^{\pm i\Omega t\frac{D_3}{|D|}}\}_{t\in \mathbb {R}}\), we prove unique existence of global mild solutions to initial value problem and mild solutions to time-periodic problem of the rotating Navier–Stokes equations under some precise conditions, respectively, which permit the initial velocity and the time-periodic external force to be arbitrarily large provided that the speed of the rotation is fast enough. These results improve the related ones obtained in Iwabuchi and Takada (Math Ann 357:727–741, 2013) and Koh et al. (Adv diff Equ 19:857–878, 2014), and the result on the time-periodic problem can also be regarded as an enhancement and complement of that in Iwabuchi and Takada (J Evol Equ 12:985–1000, 2012). Meanwhile, based on the so-called Gevrey estimates, which are motivated by the work of Foias and Temam (J Funct Anal 87:359–369, 1989), we particularly verify that the obtained mild solutions are analytic in the spatial variables.