Asymptotic behavior of a generalized Navier-Stokes-alpha model and applications to related models

TL;DR

This paper analyzes the long-term behavior of a generalized Navier-Stokes-alpha model in 3D, establishing existence, uniqueness, and stability of solutions, and exploring the structure of global attractors under various conditions.

Contribution

It introduces a unified framework for studying asymptotic behavior of generalized alpha models, including existence, uniqueness, and attractor properties, with applications to related fluid dynamics equations.

Findings

Existence of finite energy solutions for the model.

Existence of strong and weak global attractors under different conditions.

Conditions under which the attractor reduces to a unique stationary solution.

Abstract

We consider a generalized alpha-type model in the whole three-dimensional space and driven by a stationary (time-independent) external force. This model contains as particular cases some relevant equations of the fluid dynamics, among them the Navier-Stokes-Bardina's model, the critical alpha-model, the fractional and the classical Navier-Stokes equations with an additional drag/friction term. First, we study the existence and in some cases the uniqueness of finite energy solutions. Then, we use a general framework to study their long time behavior with respect to the weak and the strong topology of the phase space. When the uniqueness of solutions is known, we prove the existence of a strong global attractor. Moreover, we proof the existence of a weak global attractor in the case when the uniqueness of solutions is unknown. The weak/global attractor contains a particular kind of solutions to our model, so-called the stationary solutions. In all generality we construct these solutions, and we study their uniqueness, orbital and asymptotic stability in the case when some physical constants in our model are large enough. As a bi-product, we show that in some cases the weak/global attractor reduces down to the unique stationary solution.