The rotating periodic, spiral-like almost periodic and spiral-like almost automorphic solutions of Navier-Stokes equations with the Coriolis force
Ziying Chen, Yong Li
TL;DR
This work analyzes the Navier–Stokes equations with the Coriolis force in a rotating frame and establishes the existence and uniqueness of rotating periodic, spiral-like almost periodic, and spiral-like almost automorphic mild solutions under small external forcing. The authors develop a robust functional-analytic framework using the explicit rotating semigroup, $L^p-L^q$ bounds, and Besov-space techniques to handle the nonlinear term via a fixed-point argument in a suitable $X^{r,q}$ space. By introducing affine-periodic and spiral-like recurrence structures for the forcing, they prove that the solution inherits the same spatio-temporal patterns, broadening the scope from classical time-periodic solutions to more complex, spiral-type patterns. This provides a rigorous basis for modeling recurrent phenomena in rotating fluids, with potential applications to atmospheric and oceanic flows where spiral-like and affine symmetries arise.
Abstract
We consider the spatio-temporal periodic problem for the Navier-Stokes equations with a small external force in the rotational framework. We prove the existence and uniqueness of the rotating periodic, spiral-like almost periodic and spiral-like almost automorphic solutions of Navier-Stokes equations with the Coriolis force.