Concerning to the non-stationary Navier-Stokes flow with a nonzero constant velocity at infinity, just a few results have been obtained, while most of the results are for the flow with the zero velocity at infinity. The temporal stability of stationary solutions for the Navier-Stokes flow with a nonzero constant velocity at infinity has been studied by Enomoto and Shibata (J Math Fluid Mech 7:339-367, 2005), in L p spaces for p ≥ 3. In this article, we first extend their result to the case 3 2 < p by modifying the method in Bae and Jin (J Math Fluid Mech 10: [423][424][425][426][427][428][429][430][431][432][433] 2008) that was used to obtain weighted estimates for the Navier-Stokes flow with the zero velocity at infinity. Then, by using our generalized temporal estimates we obtain the weighted stability of stationary solutions for the Navier-Stokes flow with a nonzero velocity at infinity.Mathematics Subject Classification (2010). Primary 35Q30; Secondary 76D05.

The 2D g-Navier-Stokes equations have the following form,with the continuity equationwhere g is a smooth real valued function. We get the Navier-Stokes equations, for g = 1. In this paper, we investigate solutions {u g , p g } of the g-Navier-Stokes equations, as g → 1 in some suitable spaces.

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 g-Navier-Stokes equations in spatial dimension 2 are the following equations introuduced inwith the continuity equation 1 g r ¢ (gu) = 0:Here, we show the existence and uniqueness of solutions of g-Navier-Stokes equations on R n for n = 2;3.

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