Bulletin of the
Korean Mathematical Society BKMS

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.