Attractors for the Navier--Stokes--Voight equations and their dimension
Alexei Ilyin, Sergey Zelik
TL;DR
This work analyzes the Navier–Stokes–Voight (NSV) regularization of the Navier–Stokes system in bounded domains and on the torus, introducing the dual dimensionless parameters $(\alpha\lambda_1)$ and $G$ and establishing attractor theory for the NSV model. It obtains explicit finite-dimensional attractor bounds in 3D that scale as $\dim_F\mathscr A_\alpha \preceq (\alpha\lambda_1)^{-3/4}\cdot G^2\min[(\alpha\lambda_1)^{-3/4}, G^2]$, and in 2D as $\dim_F\mathscr A_\alpha \le \frac{(\alpha\lambda_1+1) c_{LT}}{2} G^2$, with additional results showing convergence to the classical NSE attractor when $\alpha\to0$ under Dirichlet and periodic boundaries. The analysis hinges on dissipativity in $\mathbf H^1$, linearized $n$-trace estimates, and Lieb–Thirring inequalities, with explicit constants provided. An appendix supplies a detailed bound for the auxiliary function $\rho$ using spectral estimates of the Laplacian on the torus, enabling the required control in the dimension estimates. Overall, the paper advances explicit, dimensionless attractor bounds for NSV and clarifies the $\alpha\to0$ limit in 2D.
Abstract
The Voight regularization of the Navier--Stokes system is studied in a bounded domain and on the torus. In the 3D case we obtain new explicit bounds for the attractor dimension improving the previously known results. In the 2D case we show that the estimates so obtained converge to the known estimates for the attractor of the Navier--Stokes system as the regularization parameter tends to zero both for the Dirichlet and the periodic boundary conditions.