Communications in Mathematical Sciences

Volume 21 (2023)

Number 8

Determination for the 2D incompressible Navier–Stokes equations in Lipschitz domain

Pages: 2301 – 2328

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n8.a10

Authors

Xin-Guang Yang (Department of Mathematics & Information Science, Henan Normal University, Xinxiang, Henan, China)

Meng Hu (Department of Mathematics & Information Science, Henan Normal University, Xinxiang, Henan, China)

To Fu Ma (Department of Mathematics, University of Brasília, DF, Brazil)

Jinyun Yuan (School of Computer Science & Technology, Dongguan University of Technology, Dongguan, China; and Department of Mathematics & Information Science, Henan Normal University, Xinxiang, Henan, China)

Abstract

The number of determining modes is estimated for the 2D Navier–Stokes equations subject to an inhomogeneous boundary condition in Lipschitz domains by using an appropriate set of points in the configuration space to represent the flow by virtue of the Grashof number and the measure of Lipschitz boundary based on a stream function and some delicate estimates. The asymptotic determination via finite functionals for 2D autonomous Navier–Stokes equations in Lipschitz domains has been derived for the trajectories inside global attractor with finite Hausdorff dimension, which leads to this fluid flow reducing to a functional ordinary differential equation.

Keywords

Navier–Stokes equations, Lipschitz domain, determining modes, Grashof number

2010 Mathematics Subject Classification

35B40, 35B41, 35Q30, 76D03, 76D05

Full Text (PDF format)

The authors’ work was partly supported by the Incubation Fund Project of Henan Normal University (No. 2020PL17); by the Key project of Henan Education Department (No. 22A110011); by the Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003); and by the National Natural Science Foundation of China (No. 12171087).

Received 30 May 2022

Received revised 14 February 2023

Accepted 11 April 2023

Published 15 November 2023