Small viscosity sharp estimates for the global attractor of the 2-D damped-driven Navier-Stokes equations

Abstract

Abstract. We consider in this article the damped and driven two-dimensional Navier-Stokes equations at the limit of small viscosity coefficient ν → 0 + . In particular, we obtain upper bounds of the order ν −1 on the fractal and Hausdorff dimensions of the global attractor for the system on the torus T 2 , on the sphere S 2 and in a bounded domain. Furthermore, in the case of the torus, we establish a lower bound of the order ν −1 . This sharp estimate is remarkably smaller than the well established sharp bound for the dimension of the global attractor of the Navier-Stokes equations on the torus T 2 , which is of the order ν −4/3 . This means that the damping/friction term plays a significant role in reducing the number of degrees of freedom in this two-dimensional model. This, we believe, is done by dissipating the energy at the large spatial scales which is transferred to these scales via the inverse cascade mechanism. Finally, we remark that the system of equations studied here is related to the Stommel-Charney barotropic ocean circulation model of the gulf stream.

Infinite Energy Solutions for Damped Navier–Stokes Equations in $${\mathbb{R}^2}$$

“…The mathematical theory of damped Navier-Stokes and Euler equations is of a big current interest, see CV08,CVZ11,CR07,Il91,IlT06,IMT04 [4,5,6,10,11,12] and references therein. However, most part of these papers study either the case of bounded underlying domain (e.g., with periodic boundary conditions) or the 2000 Mathematics Subject Classification.…”

“…The mathematical theory of damped Navier-Stokes and Euler equations is of a big current interest, see CV08,CVZ11,CR07,Il91,IlT06,IMT04 [4,5,6,10,11,12] and references therein. However, most part of these papers study either the case of bounded underlying domain (e.g., with periodic boundary conditions) or the case of finite energy solutions in the whole space R 2 and very few is known about the infiniteenergy solutions (for instance, starting with u 0 ∈ L ∞ (R 2 ) or from the proper uniformly local Sobolev space) which are natural from the physical point of view (for instance, for applications to the so-called uniform turbulence theory or/and statistical hydrodynamics, see VF80 [21] for the details).…”

Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity

“…We observe from equation (5.7) (and also see [22]) that for any t 0 > 0, 24) and the bound depends only on t 0 and σ. Let (u n , u tn ) be the corresponding solution of (u n 0 , v n 0 ) ∈ B σ for problem (5.7), n = 1, 2, · · · .…”

“…Especially, in the autonomous case, the pullback attractor coincides with the global attractor; see [3,35,36,40,24]. Moreover, Chepyzhov & Vishik [14] define the concept of kernel sections for non-autonomous dynamical systems, which correspond to the fibers A σ in the above Definition 3.2 of a pullback attractor.…”

On the domain of analyticity and small scales for the solutions of the damped-driven 2d Navier–Stokes equations

“…That is why in this work the friction coefficient µ > 0 will be fixed and we consider the system at the limit when ν → 0 + . Sharp estimates (as ν → 0) for the Hausdorff and the fractal dimensions of the global attractor of the system (1.6) were first obtained in the case of the squareshaped domain in [27] (γ = 1). Then the case of an elongated domain was studied in [29] (γ → 0), where it was shown that…”