$(H,H^3)$-smoothing effect and convergence of solutions of stochastic two-dimensional anisotropic Navier-Stokes equations driven by colored noise

TL;DR

This paper investigates the regularity, attractors, and convergence of solutions to stochastic anisotropic Navier-Stokes equations on a 2D torus, establishing smoothing effects and finite-dimensional attractors under colored noise.

Contribution

It introduces new regularity results and attractor properties for stochastic anisotropic Navier-Stokes equations driven by colored noise, including convergence analysis as noise intensity diminishes.

Findings

Existence of tempered $(H,H^2)$-random attractors with finite fractal dimension.

Establishment of an $H^2$-bounded absorbing set and smoothing effect.

Proven convergence of solutions as noise intensity parameter tends to zero.

Abstract

This paper is devoted to the higher regularity and convergence of solutions of anisotropic Navier-Stokes (NS) equations with additive colored noise and white noise on two-dimensional torus $\mathbb T^2$. Under the conditions that the external force $f(\textbf{x})$ belongs to the phase space $ H$ and the noise intensity function $h(\textbf{x})$ satisfies $\|\nabla h\|_{L^\infty} \leq \sqrt{\pi\delta} \frac{\nu \lambda_1}{2}$, it was proved that the random anisotropic NS equations possess a tempered $(H,H^2)$-random attractor whose (box-counting) fractal dimension in $H^2$ is finite. This was achieved by establishing, first, an $H^2$ bounded absorbing set and, second, an $(H,H^2)$-smoothing effect of the system which lifts the compactness and finite-dimensionality of the attractor in $H$ to that in $H^2$. Since the force $f$ belongs only to $H$, the $H^2$-regularity of solutions as well as the $H^2$-bounded absorbing set was constructed by an indirect approach of estimating the $H^2$-distance between the solution of the random anisotropic NS equations and that of the corresponding deterministic anisotropic NS equations. When the external force $f(\textbf{x})$ belongs to $H^2$ and the noise intensity function $h(\textbf{x})$ satisfies the Assumption 2, it was proved that the random anisotropic NS equations possess a tempered $(H,H^3)$-random attractor whose (box-counting) fractal dimension in $H^3$ is finite. Finally, we prove the upper semi-continuity of random attractors and the convergence of solutions of (8.3) as $\delta\rightarrow0$ in the spaces $(H,H)$, $(H,H^1)$, $(H^1,H^2)$ and $(H^2,H^3)$, respectively.