ON PRESSURE STABILIZATION METHOD FOR NONSTATIONARY NAVIER-STOKES EQUATIONS

. In this paper, we consider the nonstationary Navier-Stokes equations approximated by the pressure stabilization method. We can obtain the local in time existence theorem for the approximated Navier-Stokes equations. Moreover we can obtain the error estimate between the solution to the usual Navier-Stokes equations and the Navier-Stokes equations approximated by the pressure stabilization method.

1. Introduction. The mathematical description of fluid flow is given by the following Navier-Stokes equations:    ∂ t u − ∆u + (u · ∇)u + ∇π = f ∇ · u = 0 t ∈ (0, ∞), x ∈ Ω, u(0, x) = a x ∈ Ω, u(t, x) = 0 x ∈ ∂Ω, where the fluid vector fields u = u(t, x) = (u 1 (t, x), . . . , u n (t, x)) and the pressure π = π(t, x) are unknown function, the external force f = f (t, x) is a given vector function, the initial data a is a given solenoidal function and Ω is some bounded domain ( see section 2 for detail). It is well-known that analysis of Navier-Stokes equations (1) is very important in view of both mathematical analysis and engineering, however the problem concerning existence and regularity of solution to (1) is unsolved for a long time. One of the difficulty of analysis for (1) is the pressure term ∇π and incompressible condition ∇ · u = 0. In numerical analysis, some penalty methods (quasi-compressibility methods) are employed as the method to overcome this difficulty. They are methods that eliminate the pressure by using approximated incompressible condition. For example, setting α > 0 as a perturbation parameter, we use ∇ · u = −π/α in the penalty method, ∇ · u = ∆π/α in the pressure stabilization method and ∇ · u = −∂ t π/α in the pseudocompressible method. In this paper, we consider the Navier-Stokes equations with incompressible condition approximated by pressure stabilization method. Namely we consider the following equations: x ∈ Ω, u α (t, x) = 0, ∂ n π α (t, x) = 0 x ∈ ∂Ω. (2) (2) may be considered as a singular perturbation of (1). As α → ∞, (2) tends to (1) formally and we cancel the Neumann boundary condition for the pressure. Pressure stabilization method was first introduced by Brezzi and Pitkäranta [1]. They considered the approximated stationary Stokes equations which are linearized Navier-Stokes equations with the approximated incompressible condition ∇ · u α = ∆π α /α. They obtained the following error estimate by using the energy methods: Nazarov and Specovius-Neugebauer [7] considered the same approximate Stokes problem and derived asymptotically precise estimates for solution to the approximated problem as α → ∞ by using the parameter-dependent Sobolev norms. Their results are not available by the usually applied energy methods and are optimal results. These results introduced above are concerning the stationary Stokes equations and there are few results concerning the nonstationary Stokes equations and Navier-Stokes equations. As far as the authors know, only the result due to Prohl [8] is known as the results concerning the nonstationary problem. In [8], Prohl considered the sharp a priori estimate for the pressure stabilization method under some assumptions and showed the following error estimates: where τ = τ (t) = min(t, 1). Since their results are proved based on energy method, all of these estimates are in L 2 framework for the space. In this paper, we shall use the maximal regularity theorem in order to prove the local in time existence theorem and the error estimate in the L p in time and the L q in space framework with n/2 < q < ∞ and max{1, n/q} < p < ∞. Here, the maximal regularity theorem means that each term in the abstract Cauchy problem is well-defined and has the same regularity. To be precisely, when we consider the Cauchy problem where X be a Banach space, A be closed linear unbounded operator in X with dense domain D(A) and f : R + → X is a given function has the maximal regularity, the maximal regularity theorem means for each f ∈ L p (R + , X) there exists a unique solution u to (4) almost everywhere and satisfying ∂ t u, Au ∈ L p (R + , X). However it is difficult to analyze equations (2) as it is by using the maximal regularity theorem because the regularity of solution to the first equation is different from the one of the second equations in (2). For this purpose, in order to adjust the regularity of the solution to their equations, we consider the following equations instead of approximated incompressible conditions in (2): for 1 < q < ∞. We notice that (5) is a weak form of the approximated incompressible condition ∇ · u α = α −1 ∆u α . We call (5) approximated weak incompressible condition in this paper. Therefore we consider under the approximated weak incompressible condition (5) in L q -framework (n/2 < q < ∞). This paper consists of the following five sections. In section 2, we present the main results on local in time unique existence of solution to (6) and certain error estimate between the solutions to (6) and (1) under the weak incompressible condition (Theorem 2.1 and Theorem 2.12). Following the argument due to Shibata and Kubo [10], we can prove the main results by contraction mapping principle with the help of the maximal L p -L q regularity theorem. After stating the main results, we present the maximal L p -L q regularity theorem for linearized (6) (Theorem 2.2 and Theorem 2.10 ) and the theorem concerning the existence of R-bounded solution operator for linearized problem (Theorem 2.7). As was seen in Shibata and Shimizu [11], the maximal L p -L q regularity theorem is direct consequence of Theorem 2.7 concerning the generalized resolvent problem for the linearized equations with the help of Weis' operator valued Fourier multiplier theorem (Theorem 2.6), so that the main part of this paper is to show Theorem 2.7. Moreover another consequence of Theorem 2.7 is the resolvent estimate (Corollary 2.8), which implies the construct of the semigroup T α (t) for linearized (6). By real interpolation, we obtain some estimates for T α (t) (Theorem 2.9 and Theorem 2.11). In section 3, as preliminary, we shall introduce some theorems and lemmas which play important role in this paper. In section 4, we consider the generalized resolvent problem for linearized problem in some bounded domain. For this purpose, we first consider the problem in the whole space case and the half-space case. By using the change of variable with their results, we shall prove the generalized bounded domain cases. In section 5, the following the argument due to Shibata and Kubo [10], we show the local in time existence theorem for (6) and prove the error estimates (Theorem 2.1 and Theorem 2.12.).
2. Main results. Before we describe main theorem, we shall introduce some functional spaces and notations throughout this paper. The letter C denotes generic constants and the constant C a,b,... depends on a, b, . . . . The values of constants C and C a,b,... may change from line to line. For 1 < q < ∞, let q = q/(q − 1). For any two Banach spaces X and Y , L(X, Y ) denotes the set of all bounded linear operators from X into Y and we write L(X) = L(X, X) for short. Hol(U, X) denotes the set of all X-valued holomorphic functions defined on a complex domain U . As the complex domain where a resolvent parameter belongs, we use Σ ε = {λ ∈ C\{0} | | arg λ| < π − ε} and Σ ε,λ0 = {λ ∈ Σ ε | |λ| ≥ λ 0 } for 0 < ε < π/2 and λ 0 > 0. For any domain D, Banach space X and 1 ≤ q ≤ ∞, L q (D, X) denotes the usual Lebesgue space of X-valued functions defined on D and · Lq(D,X) denotes its norm. We use the notation L q (D) = L q (D, R), · Lq(D) = · Lq(D,R) and for a, b, . . . , c ∈ L q (D), (a, b, . . . , c) Lq(D) = a Lq(D) + b Lq(D) + · · · + c Lq(D) . In a similar way, for 1 ≤ q ≤ ∞ and a positive integer m, W m q (D, X) denotes the Sobolev spaces of X-valued functions of defined on D. We often use the same symbols for denoting the vector and scalar function spaces if there is no confusion.

TAKAYUKI KUBO AND RANMARU MATSUI
For a Banach space X and some γ 0 ∈ R, we set In order to deal with the pressure term, we use the following functional spaces: Since our proof is based on Fourier analysis, we next introduce the Fourier transform and the Laplace transform. We define the Fourier transform, its inverse Fourier transform, the Laplace transform and its inverse Laplace transform bŷ , respectively, where x, ξ ∈ R n , λ = γ + iτ ∈ C and x · ξ is usual inner product: x · ξ = n j=1 x j ξ j . Furthermore, we define the Fourier-Laplace transform by By using Fourier transform and Laplace transform, we define H s p,γ0 (R, X) for a Banach space X. For λ = γ + iτ , we define the operator Λ s γ as . For 0 < s < 1 and γ 0 > 0, we define the space H s p,γ0 (R, X) as H s p,γ0 (R, X) = {f ∈ L p,γ0 (R, X) | e −γt Λ s γ f Lp(R,X) < ∞(∀γ ≥ γ 0 )}. In this paper, we assume next assumption for our domain Ω. Assumption 2.1. Let n/2 < q < ∞ and n < r < ∞. Let Ω be a uniform W 2−1/r r domain introduced in [5] and L q (Ω) has the Helmholtz decomposition.
Here the assumption on a uniformly W 2−1/r r domain is used when we reduce the problem on the bounded domain to one on the bent half-space and on the whole space (see section 4.3 for detail). According to Galdi [6], that "L q (Ω) has the Helmholtz decomposition" is equivalent that the following weak Neumann problem is uniquely solvable: for f ∈ L q (Ω), (∇θ, ∇ϕ) = (f, ∇ϕ) ϕ ∈ W 1 q (Ω). The map P Ω and Q Ω are defined by Q Ω f = θ, where θ is the solution to the above weak Neumann problem and P Ω f = f − ∇Q Ω f . P Ω is called the Helmholtz projection. We remark that if q = 2, L 2 (Ω) has the Helmholtz decomposition for any Ω (see Galdi [6]).
First main result is concerned with the local in time existence theorem for (6) with approximated weak incompressible condition (5).
Theorem 2.1. Let n ≥ 2, n/2 < q < ∞ and max{1, n/q} < p < ∞. Let α > 0 and T 0 ∈ (0, ∞). For any M > 0, assume that the initial data a α ∈ B (Ω) and the external force f ∈ L p ((0, T 0 ), L q (Ω) n ) satisfy Then, there exists T * ∈ (0, T 0 ) depending on only M such that (6) under (5) has a unique solution (u α , π α ) of the following class: . Moreover the following estimate holds: Here we state the outline of the proof of main theorem (Theorem 2.1). We show Theorem 2.1 by using the contraction mapping principle with two type maximal regularity theorems (Theorem 2.2 and Theorem 2.9). In order to prove Theorem 2.2, we use the Weis' operator valued Fourier multiplier theorem. For this purpose, we have to show the existence of R-bounded solution operator to the generalized resolvent problem of (6) (see Theorem 2.7 for detail). In order to prove Theorem 2.9, we need the some estimate of semigroup T α (t) for linearized problem of (6). For this purpose, we have to show the resolvent estimate (Corollary 2.8), which is a corollary of Theorem 2.7. Therefore our main task is to show Theorem 2.7.
In order to prove Theorem 2.2, we use the operator valued Fourier multiplier theorem due to Weis [12]. This theorem needs R-boundedness of solution operator. To this end, we first introduce the definition of R-boundedness.
. . , N ) and for all sequences {γ j (u)} N j=1 of independent, symmetric, {−1, 1}-valued random variables on [0,1], there holds the inequality: The smallest such C is called R-bound of T on L(X, Y ), which is denoted by R(T ).
Remark 2.5. According to [3], the following properties concerning R-boundedness is known. From Definition 2.4, R-boundedness of the family of operators implies uniform boundedness.
Moreover it is well-known that R-bounds behave like norms. Namely, the following properties hold.
The following theorem is the operator valued Fourier multiplier theorem proved by Weis [5] for X = Y = L q (Ω).
Theorem 2.11. Let 1 < p, q < ∞ and α > 0. Let λ 0 be a number obtained in Theorem 2.7. For a E ∈ B for any T > 0.
By above two theorems, we can obtain the following theorem concerned with the error estimates.
3. Preliminary. In this section, we shall introduce some lemmas and definitions, which plays important role for our proof. Before we describe some propositions and lemmas, we introduce the notation of symbols. Set where ξ = (ξ 1 , . . . , ξ n−1 ). Here E(ω λ ) is the symbol corresponding to heat equation and M(ω λ , r, x n ) is the symbol corresponding to Stokes equations. We next introduce some lemmas. In order to apply the operator-valued Fourier multiplier theorem proved by Weis [12], we need the R-boundedness of solution operator to (8). However since it is difficult to prove R-boundedness directly from its definition, we first introduce the following sufficient condition for showing Rboundedness of solution operator given in Theorem 3.3 in Enomoto and Shibata [4].
This lemma is proved in a similar way to Lemma 5.4 in [11] with the following lemma. Lemma 3.3. For 0 < ε < π/2, let λ ∈ Σ ε . (i) There exist positive constants C 1 , C 2 and C 3 depending on ε such that the following inequalities hold: (ii) There exist positive constants C such that the following inequalities hold: for any s ∈ R and multi-index δ.

4.
Rboundedness of the solution operator to resolvent problem. Goal of this section is to prove the R-boundedness of the solution operator to the following resolvent problem (10) in Ω: where λ ∈ Σ ε,λ0 (0 < ε < π/2, λ 0 > 0) under the approximated weak incompressible condition (9). Our method is based on cut-off technique. For this purpose, we shall first prove the whole space case. Secondly we shall prove the half-space case by using the result for the whole space case and some lemma introduced in section 3. Next we shall prove the bent half-space case by reducing to the result for the half-space case with the change of variable. Finally we shall prove the bounded domain case by using the result for the whole space and the bent half-space case with cut-off technique.
Proof. In order to prove the R-boundedness of solution operator by using Theorem 3.1, we shall obtain the solution formula to (10) under (9) by using Fourier transform. By the property of Helmholtz projection, we know ∇π α = α∇Q R n (u α − g) and F[∇Q R n v] = |ξ| −2 ξ(ξ · v). Applying the Fourier transform to (10), we obtain the following solution formula : u α,j (x) = u j (x)+u E α,j (x) and π α (x) = π(x)+π E α (x), where (u, π) is the solution to Stokes equations given by for j = 1, . . . , n and the error term (u E α , π E α ) given by for j = 1, . . . , n. Since in the whole space case, it is well-known that the solution operator to Stokes equations is R-bounded ( [11] for detail), we consider the only error term (u E α , π E α ). By Leibniz rule, for = 0, 1, we obtain which implies from Theorem 3.1 This completes the proof of Theorem 4.1.
In order to prove Theorem 4.3 by Lemma 3.2, we shall obtain the solution formula to (10) under (9). By density argument, we may let f, g ∈ C ∞ 0 (R n + ). In this case, equation (10) under (9) is equivalent to the following equations: We shall obtain the solution formula to (26). For this purpose, we extend the external force f and g to the whole space. For f = (f 1 , . . . , f n ) and g = (g 1 , . . . , g n ), let F = (f e 1 , . . . , f e n−1 , f o n ) and G = (g e 1 , . . . , g e n−1 , g o n ), where , where x = (x 1 , . . . , x n−1 ). We consider the resolvent problem with F and G: Here we remark that from the definition of our extension, (U α , Π α ) enjoys the boundary condition By the result for the whole space and the definition of our extension, the following estimates hold: Setting u α = w α + U α and π α = ρ α + Π α , we see that to solve (26) is equivalent to solve where h j = −(U α ) j for j = 1, . . . , n − 1 and h n = 0. Applying div and (λ + α − ∆)∆ to the first equation in (30), we obtain By applying the partial Fourier transform defined by to (30) and (31) , we have and where iξ · w α = n−1 j=1 (iξ j )( w α ) j . Since from (33), we see the solution ( w α , ρ α ) can be expressed by for j = 1, . . . , n, we shall find the solution to (32) having the form (34). By substituting (34) to (32), we see Therefore, we obtain the solution formula ( Since the symbol M(a, b, x n ) defined by (16) has the following properties: and by g(0) = − ∞ 0 ∂ n g(y n )dy n , we have where E(z) is defined by (16). Therefore, settingξ j = ξ j /r, we obtain + M(ω λ , r, x n + y n )(r − D n )r h k (ξ , y n ))](x )dy n , We remark that (w, ρ) is the solution to the usual Stokes equations and (w E , ρ E ) is the error between the solution to Stokes equations and Stokes equations approximated by pressure stabilization. Since Shibata and Shimizu [11] proved Rboundedness of solution operator to Stokes equations, it is sufficient to consider (w E α , ρ E α ) only. For this purpose, we prepare the following lemma.
By using the cut-off technique with Theorem 4.5, we shall prove Theorem 2.7.
Proof of Theorem 2.12. Let (u * , π * ) be a solution to (8) with f = g = 0 and a α = a E . By Theorem 2.9, the following estimates hold.
In a similar way to Theorem 2.1, taking T sufficiently small if necessary, we can prove that Φ is the contraction mapping on X T ,L E . Therefore we obtain Theorem 2.12.