Global solvability of the rotating Navier-Stokes equations with fractional Laplacian in a periodic domain

Abstract

We consider existence of global solutions to equations for three-dimensional rotating fluids in a periodic frame provided by a sufficiently large Coriolis force. The Coriolis force appears in almost all of the models of meteorology and geophysics dealing with large-scale phenomena. In the spatially decaying case, Koh et al. (J Differ Equ 256:707–744, 2014) showed existence for the large times of solutions of the rotating Euler equations provided by the large Coriolis force. In this case the resonant equation does not appear anymore. In the periodic case, however, the resonant equation appears, and thus the main subject in this case is to show existence of global solutions to the resonant equation. Research in this direction was initiated by Babin et al. (Indiana Univ Math J 48:1133–1176, 1999) who treated the rotating Navier-Stokes equations on general periodic domains. On the other hand, Golse et al. (Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains. Instability in Models Connected with Fluid Flows. I. Springer, New York, pp 301–338. arXiv:0704.0337v1 , 2008) considered bursting dynamics of the resonant equation in the case of a cylinder with no viscosity. Thus we may not expect to show global existence of solutions to the resonant equation without viscosity in the periodic case. In this paper we show existence of global solutions for fractional Laplacian case (with its power strictly less than the usual Laplacian) in the periodic domain with the same period in each direction. The main ingredient is an improved estimate on resonant three-wave interactions, which is based on a combinatorial argument.