The dimension of attractor of the 2D g-Navier–Stokes equations

Abstract

The g-Navier-Stokes equations in spatial dimension 2 were introduced by Roh aswith the continuity equationwhere g is a suitable smooth real valued function. Roh proved the existence of global solutions and the global attractor, for the spatial periodic and Dirichlet boundary conditions. Roh also proved that the global attractor A g of the g-Navier-Stokes equations converges (in the sense of upper continuity) to the global attractor A 1 of the Navier-Stokes equations as g → 1 in the proper sense.In this paper, we will estimate the dimension of the global attractor A g , for the spatial periodic and Dirichlet boundary conditions. Then, we will see that the upper bounds for the dimension of 437 the global attractors A g converge to the corresponding upper bounds for the global attractor A 1 as g → 1 in the proper sense.  2005 Elsevier Inc. All rights reserved.

The Finite‐Dimensional Uniform Attractors for the Nonautonomous g‐Navier‐Stokes Equations

“…the constant c 1 , c 2 of (3.29) and (3.32) of Chapter V I in [15] and [2], λ 1 is the first eigenvalue of the Stokes operator and…”

“…where g g x 1 , x 2 is a suitable smooth real-valued function defined on x 1 , x 2 ∈ Ω and Ω is a suitable bounded domain in R 2 . Notice that if g x 1 , x 2 1, then 1.1 reduce to the standard Navier-Stokes equations.…”

ASYMPTOTIC BEHAVIOR OF STRONG SOLUTIONS TO 2D g-NAVIER-STOKES EQUATIONS

“…In the last few years, the existence and long-time behavior of weak solutions to 2D g-Navier-Stokes equations have been studied extensively in both autonomous and non-autonomous cases (see e.g. [1,4,5,6,7,8,10,14]). However, to the best of our knowledge, little seems to be known about strong solutions of the 2D g-Navier-Stokes equations.…”

Spectral Galerkin Method in Space and Time for the 2Dg-Navier-Stokes Equations

“…In the last few years, the existence and long-time behavior of both weak and strong solutions to the 2D -Navier-Stokes equations have been studied extensively (cf. [2][3][4][5][6][7][8][9]). In this paper, we aim to study numerical approximation of the strong solutions to problem (1).…”