Bull. Korean Math. Soc. 2021; 58(5): 1109-1127
Online first article August 26, 2021 Printed September 30, 2021
https://doi.org/10.4134/BKMS.b200470
Copyright © The Korean Mathematical Society.
Cung The Anh, Le Thi Thuy, Le Tran Tinh
Hanoi National University of Education; Electric Power University; Hong Duc University
We study the long-term dynamics for a family of incompressible three-dimensional Leray-$\alpha$-like models that employ the spectral fractional Laplacian operators. This family of equations interpolates between incompressible hyperviscous Navier-Stokes equations and the Leray-$\alpha$ model when varying two nonnegative parameters $\theta_1$ and $\theta_2$. We prove the existence of a finite-dimensional global attractor for the continuous semigroup associated to these models. We also show that an operator which projects the weak solution of Leray-$\alpha$-like models into a finite-dimensional space is determining if it annihilates the difference of two ``nearby" weak solutions asymptotically, and if it satisfies an approximation inequality.
Keywords: Leray-$\alpha$-like models, fractional Laplacian, weak solution, global attractor, fractal dimension, asymptotic determining operator
MSC numbers: 35Q35, 37L30, 76D03, 76F20, 76F65
Supported by: This work is supported by Vietnam Ministry of Education and Training under grant number B2021-SPH-15.
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