Dynamics for Generalized Incompressible Navier--Stokes Equations in ℝ2

Abstract

Abstract In this paper, we consider the dynamics for damped generalized incompressible Navier–Stokes equations defined on ℝ 2 ${\mathbb{R}^{2}}$ . The generalized Navier–Stokes equations here refer to the equations obtained by replacing the Laplacian in the classical Navier–Stokes equations by the more general operator ( - Δ ) α ${(-\Delta)^{\alpha}}$ with α ∈ ( 1 2 , 1 ) ${\alpha\in(\frac{1}{2},1)}$ . We prove that the rate of dissipation of enstrophy vanishes as ν → 0 + ${\nu\to 0^{+}}$ , where ν is the viscosity parameter. Moreover, we prove the existence and finite dimensionality of a global attractor in ( H 1 ⁢ ( ℝ 2 ) ) 2 ${(H^{1}(\mathbb{R}^{2}))^{2}}$ as ν > 0 ${\nu>0}$ is kept fixed for the generalized Navier–Stokes equations.