In this paper, we first prove the global existence of strong solutions to 3-D incompressible Navier-Stokes equations with solenoidal initial data, which writes in the cylindrical coordinates is of the form: A(r,z)\cosN\theta+B(r,z)\sinN\theta, provided that N is large enough. In particular, we prove that the corresponding solution has almost the same frequency N for any positive time. The main idea of the proof is first to write the solution in trigonometrical series in \theta variable and estimate the coefficients separately in some scale-invariant spaces, then we handle a sort of weighted sum of these norms of the coefficients in order to close the a priori estimate of the solution. Furthermore, we shall extend the above well-posedness result for initial data which is a linear combination of axisymmetric data without swirl and infinitely many large mode trigonometric series in the angular variable. 

Publication: 

Advances in Mathematics, Volume 438, February 2024, 109475 

http://dx.doi.org/10.1016/j.aim.2023.109475 

Author: 

Yanlin Liu 

School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, MOE, Beijing Normal University, 100875 Beijing, China 

Ping Zhang 

Academy of Mathematics & Systems Science and Hua Loo-Keng Center for Mathematical Sciences, Chinese Academy of Sciences, Beijing 100190, China 

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 

Email: zp@amss.ac.cn