Quasi-periodic solutions for the incompressible Navier-Stokes equations with nonlocal diffusion-Reference-Cited by-同舟云学术

Quasi-periodic solutions for the incompressible Navier-Stokes equations with nonlocal diffusion

Published:2023 Issue:12 Volume:31 Page:7182-7194
ISSN:2688-1594
Container-title:Electronic Research Archive
language:
Short-container-title:era

Author:

Ji Shuguan,Li Yanshuo

Abstract

<abstract><p>This paper studied the incompressible Navier-Stokes (NS) equations with nonlocal diffusion on $ \mathbb{T}^d (d \ge 2) $. Driven by a time quasi-periodic force, the existence of time quasi-periodic solutions in the Sobolev space was established. The proof was based on the decomposition of the unknowns into the spatial average part and spatial oscillating one. The former were sought under the Diophantine non-resonance assumption, and the latter by the contraction mapping principle. Moreover, by constructing suitable time weighted function space and using the Banach fixed point theorem, the asymptotic stability of quasi-periodic solutions and the exponential decay of perturbation were proved.</p></abstract>

Publisher

American Institute of Mathematical Sciences (AIMS)

Subject

General Mathematics

Reference29 articles.

1. J. R. Mercado, E. P. Guido, A. J. Sánchez-Sesma, M. Íñiguez, A. González, Analysis of the Blasius's formula and the Navier-Stokes fractional equation, in Fluid Dynamics in Physics, Engineering and Environmental Applications, Springer, Berlin, Heidelberg, (2013), 475–480. https://doi.org/10.1007/978-3-642-27723-8_44

2. X. C. Zhang, Stochastic Lagrangian particle approach to fractal Navier-Stokes equations, Commun. Math. Phys., 311 (2012), 133–155. https://doi.org/10.1007/s00220-012-1414-2

3. W. A. Woyczyński, Lévy processes in the physical sciences, in Lévy Processes, Birkhäuser, Boston, MA, (2001), 241–266. https://doi.org/10.1007/978-1-4612-0197-7_11

4. J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non-Linéaires (French), Dunod, Paris, 1969.

5. J. C. Mattingly, Y. G. Sinai, An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equations, Commun. Contemp. Math., 1 (1999), 497–516. https://doi.org/10.1142/S0219199799000183