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

and Applied Analysis 3 We now define the trilinear form b by

The -Navier-Stokes equations are a variation of the standard Navier-Stokes equations.More precisely, when  ≡ const we get the usual Navier-Stokes equations.The 2D -Navier-Stokes equations arise in a natural way when we study the standard 3D problem in thin domains.We refer the reader to [1] for a derivation of the 2D -Navier-Stokes equations from the 3D Navier-Stokes equations and a relationship between them.As mentioned in [1], good properties of the 2D -Navier-Stokes equations can lead to an initiate of the study of the Navier-Stokes equations on the thin threedimensional domain Ω  = Ω × (0, ).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).To do this, we assume that (G)  ∈  1,∞ (Ω) such that where  1 > 0 is the first eigenvalue of the -Stokes operator in Ω (i.e., the operator  defined in Section 2.1 below); (F)  ∈  1,∞ (R + ;   ); that is, ,   ∈  ∞ (R + ;   ).
In this paper, in order to study the numerical approximation of strong solutions to the 2D -Navier-Stokes equations we will use the spectral Galerkin method in space and time, which is based on the eigen-subspaces of the -Stokes operator.As mentioned in [10] for the Navier-Stokes equations, this method enables us to avoid solving the fully nonlinear -Navier-Stokes equations on the low-frequency subspace, whereas to obtain the low-frequency component of the numerical solution, the usual multilevel spectral methods where Δ > 0 is the time step size and () ,   = Δ.
Here, the linear term is treated implicitly to avoid serve time step limitations, whereas the nonlinear term is kept explicitly so that the corresponding discrete system is easily invertible.It is well known that this type of scheme is only stable under some restriction on the time step size.We will obtain  2 -stability uniform in time stated in Theorem 13, provided that the following condition holds for some positive constant  depending on the data ( 0 , ], , Ω).As mentioned in [11] for the case of 2D Navier-Stokes equations, the stability condition ( 6) is a significant improvement compared with the results provided by the nonlinear Galerkin method [12] and the multilevel method [13,14].We also derive an  2 -error estimate of the numerical solution    under the stability condition (6): where   = sup ≥0 |()|, () = min{1, } and K denotes a general positive constant depending only on the data (], Ω, |∇| ∞ ,  1 ,   , ‖ 0 ‖).Noting that  −1 (  ) is a singular factor near  = 0. Compared to He's works [11] on the spectral method of the 2D Navier-Stokes equations, here we have to address some additional difficulties.Firstly, to treat the more general condition ∇ ⋅ () = 0, instead of the usual function spaces used for the Navier-Stokes equations, we use the function spaces   ,   which are defined suitably for the -Navier-Stokes equations (see Section 2.1 for details).Secondly, we have to deal with the term  in the equation, which only appears for the -Navier-Stokes equations.It is worthy noticing that when  ≡ 1, we of course recover the results for the Navier-Stokes equations in [11].
The paper is organized as follows.In the next section, we recall some results on function spaces and inequalities for the nonlinear terms related to the -Navier-Stokes equations, and some discrete Gronwall inequalities are frequently used later.In Section 3, we prove several estimates for the strong solution and the Galerkin approximate solutions of problem (1).In Section 4, we study the error analysis of the spectral Galerkin method in space.Stability and error analysis of the spectral Galerkin method in space and time are discussed in the last section.
where  0 are appropriate constants depending only on Ω.

Existence and Some Estimates of Strong Solutions
In this section, we will prove some estimates for the strong solution  and the Galerkin approximate solutions   of problem (1).First, with the operators defined in Section 2.1, one can write this problem as follows: Definition 7.For  0 ∈   given, a strong solution of problem ( 1) is a function  ∈  2 (0, ; ())∩ ([0, ];   ) for all  > 0 such that (0) =  0 , and  satisfies (35) in   for a.e. > 0.

Spectral Galerkin Method in Space
For a fixed integer , let   be the orthogonal projection of   onto   = span{ 1 , . . .,   } and   =  −   .Then, every solution  of problem (1) can be decomposed uniquely into Now, we apply   and   to (35) to obtain and the initial conditions (0) =    0 , (0) =    0 .
Using Theorem 8 and the property of   , we arrive at the following estimates of () =   (): We now define the spectral Galerkin method as follows: find   () ∈   such that with the initial condition   (0) =    0 .
In order to give an analysis of the error  −   in the  2 -norm, we begin with a technical result concerning a dual linearized -Navier-Stokes problem which is a similar problem to that used in [17].We consider, for any given  > 0 and  ∈  2 (0, ;   ), the dual problem: find for all V ∈   with Φ() = 0.It is easy to see that ( 67) is a wellposed problem and has a unique solution Φ ∈  ∞ (0, ;   ) ∩  2 (0, ; ()).
Next, we prove a regularity result of problem (67).
Finally, by combining Lemma 11 with (64) and using Theorem 8, we get the following error estimate.Theorem 12.If  ∈   , then the error () −   () satisfies the following bound:

Stability Analysis.
In this subsection, we consider the semi-implicit Euler scheme applied to the spatially discrete spectral Galerkin approximation, show the stability of this scheme, and establish some preliminaries related to the error analysis uniform in time.
We consider the semi-implicit Euler scheme and define recursively a solution {   } ⊂   such that for  ≥ 0 with the initial condition  0  =    0 , where Δ > 0 is a time step such that  0 Δ = 1 for some integer  0 and In order to derive the  2 -bound on the error   (  ) −    , we will begin with a time discrete duality argument which is similar to the one used in [11,17].We consider the dual scheme correponding to scheme (106): for any fixed  ≥ 1 and with an initial condition Φ  = 0.
The following theorem provides the  2 -stability of scheme (106).

Error Analysis.
In this subsection, we will establish the  1 -and  2 -error estimates uniform in time for the fully discrete spectral Galerkin method with the explicit time discretization for the nonlinear term.
with  0 = 0 and To derive a bound on   , we need to provide the following estimates of   .