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

The g-Navier–Stokes equations in spatial dimension 2 were introduced by Roh as∂u∂t−νΔu+(u⋅∇)u+∇p=f, with the continuity equation∇⋅(gu)=0,∇⋅(gu)=0, where 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 AgAg of the g  -Navier–Stokes equations converges (in the sense of upper continuity) to the global attractor A1A1 of the Navier–Stokes equations as g→1g→1 in the proper sense.In this paper, we will estimate the dimension of the global attractor AgAg, for the spatial periodic and Dirichlet boundary conditions. Then, we will see that the upper bounds for the dimension of the global attractors AgAg converge to the corresponding upper bounds for the global attractor A1A1 as g→1g→1 in the proper sense.