The Green function for the Stokes system with measurable coefficients

We study the Green function for the stationary Stokes system with bounded measurable coefficients in a bounded Lipschitz domain $\Omega\subset \mathbb{R}^n$, $n\ge 3$. We construct the Green function in $\Omega$ under the condition $(\bf{A1})$ that weak solutions of the system enjoy interior H\"older continuity. We also prove that $(\bf{A1})$ holds, for example, when the coefficients are $\mathrm{VMO}$. Moreover, we obtain the global pointwise estimate for the Green function under the additional assumption $(\bf{A2})$ that weak solutions of Dirichlet problems are locally bounded up to the boundary of the domain. By proving a priori $L^q$-estimates for Stokes systems with $\mathrm{BMO}$ coefficients on a Reifenberg domain, we verify that $(\bf{A2})$ is satisfied when the coefficients are $\mathrm{VMO}$ and $\Omega$ is a bounded $C^1$ domain.

(Communicated by Hongjie Dong) Abstract. We study the Green function for the stationary Stokes system with bounded measurable coefficients in a bounded Lipschitz domain Ω ⊂ R n , n ≥ 3. We construct the Green function in Ω under the condition (A1) that weak solutions of the system enjoy interior Hölder continuity. We also prove that (A1) holds, for example, when the coefficients are VMO. Moreover, we obtain the global pointwise estimate for the Green function under the additional assumption (A2) that weak solutions of Dirichlet problems are locally bounded up to the boundary of the domain. By proving a priori L q -estimates for Stokes systems with BMO coefficients on a Reifenberg domain, we verify that (A2) is satisfied when the coefficients are VMO and Ω is a bounded C 1 domain.
1. Introduction. We consider the Dirichlet boundary value problem for the stationary Stokes system      L u + Dp = f + D α f α in Ω, div u = g in Ω, u = 0 on ∂Ω, where Ω is a domain in R n . Here, L is an elliptic operator of the form where the coefficients A αβ = A αβ (x) are n × n matrix valued functions on R n with entries a ij αβ that satisfying the strong ellipticity condition; i.e., there is a constant λ ∈ (0, 1] such that for any x ∈ R n and ξ, η ∈ R n×n , we have We do not assume that the coefficients A αβ are symmetric. The adjoint operator L * of L is given by L * u = −D α (A βα (x) tr D β u). We remark that the coefficients of L * also satisfy (1.2) with the same constant λ. There has been some interest in studying boundary value problems for Stokes systems with bounded coefficients; see, for instance, Giaquinta-Modica [14]. They obtained various interior and boundary estimates for both linear and nonlinear systems of the type of the stationary Navier-Stokes system.
Our first focus is to study of the Green function for the Stokes system with L ∞ coefficients in a bounded Lipschitz domain Ω ⊂ R n , n ≥ 3. More precisely, we consider a pair (G(x, y), Π(x, y)), where G(x, y) is an n × n matrix valued function and Π(x, y) is an n × 1 vector valued function on Ω × Ω, satisfying      L x G(x, y) + D x Π(x, y) = δ y (x)I in Ω, div x G(x, y) = 0 in Ω, G(x, y) = 0 on ∂Ω.
Here, δ y (·) is Dirac delta function concentrated at y and I is the n × n identity matrix. See Definition 2.1 for the precise definition of the Green function. We prove that if weak solutions of either L u + Dp = 0, div u = 0 in B R or L * u + Dp = 0, div u = 0 in B R satisfy the following De Giorgi-Moser-Nash type estimate [u] C µ (B R/2 ) ≤ CR −n/2−µ u L 2 (B R ) , (1.3) then the Green function (G(x, y), Π(x, y)) exists and satisfies a natural growth estimate near the pole; see Theorem 2.3. It can be shown, for example, that if the coefficients of L belong to the class of VMO (vanishing mean oscillations), then the interior Hölder estimate (1.3) above holds; see Theorem 2.5. Also, we are interested in the following global pointwise estimate for the Green function: there exists a positive constant C such that |G(x, y)| ≤ C|x − y| 2−n , ∀x, y ∈ Ω, x = y. (1.4) If we assume further that the operator L has the property that the weak solution of      L u + Dp = f in Ω, div u = g in Ω, is locally bounded up to the boundary, then we obtain the pointwise estimate (1.4) of the Green function. This local boundedness condition (A2) is satisfied when the coefficients of L belong to the class of VMO and Ω is a bounded C 1 domain. To see this, we employ the standard localization method and the global L q -estimate for the Stokes system with Dirichlet boundary condition, which is our second focus in this paper. Green functions for the linear equation and system have been studied by many authors. In [21], Littman-Stampacchia-Weinberger obtained the pointwise estimate of the Green function for elliptic equation. Grüter-Widman [15] proved existence and uniqueness of the Green function for elliptic equation, and the corresponding results for elliptic systems with continuous coefficients were obtained in [6,10]. Hofmann-Kim proved the existence of Green functions for elliptic systems with variable coefficients on any open domain. Their methods are general enough to allow the coefficients to be VMO. For more details, see [16]. We also refer the reader to [18,23] and references therein for the study of Green functions for elliptic systems. Regarding the study of the Green function for the Stokes system with the Laplace operator, we refer the reader to [4,18]. In those papers, the authors obtained the global pointwise estimate (1.4) for the Green function on a three dimensional Lipschitz domain. Mitrea-Mitrea [23] established regularity properties of the Green function for the Stokes system with Dirichlet boundary condition in a two or three dimensional Lipschitz domain. Recent progress may be found in the article of Ott-Kim-Brown [25]. This work includes a construction of the Green function with mixed boundary value problem for the Stokes system in two dimensions.
Our second focus in this paper is the global L q -estimates for the Stokes systems of divergence form with the Dirichlet boundary condition. As mentioned earlier, the L q -estimate for the Stokes system is the key ingredient in establishing the global pointwise estimate for the Green function. Moreover, the study of the regularity of solutions to the Stokes system plays an essential role in the mathematical theory of viscous fluid flows governed by the Navier-Stokes system. For this reason, the L q -estimate for the Stokes system with the Laplace operator was discussed in many papers. We refer the reader to Galdi-Simader-Sohr [11], Maz'ya-Rossmann [22], and references therein. Recently, estimates in Besov spaces for the Stokes system are obtained by Mitrea-Wright [24]. In this paper, we consider the L q -estimates for Stokes systems with variable coefficients in non-smooth domains. More precisely, we prove that if the coefficients of L have small bounded mean oscillations on a Reifenberg flat domain Ω, then the solution (u, p) of the problem (1.1) satisfies the following L q -estimate: Moreover, we obtain the solvability in Sobolev space for the systems on a bounded Lipschitz domain. It has been studied by many authors that the L q -estimates for elliptic and parabolic systems with variable coefficients on a Reifenberg flat domain. We refer the reader to Dong-Kim [8,9] and Byun-Wang [3]. In particular, in [8], the authors proved L q -estimates for divergence form higher order systems with partially BMO coefficients on a Reifenberg flat domain. Their argument is based on mean oscillation estimates and L ∞ -estimates combined with the measure theory on the "crawling of ink spots" which can be found in [20]. We mainly follow the arguments in [8], but the technical details are different due to the pressure term p. The presence of the pressure term p makes the argument more involved.
The organization of the paper is as follows. In Section 2, we introduce some notation and state our main theorems, including the existence and global pointwise estimates for Green functions, and their proofs are presented in Section 4. Section 5 is devoted to the study of the L q -estimate for the Stokes system with the Dirichlet boundary condition. In Appendix, we provide some technical lemmas.
2. Main results. Before we state our main theorems, we introduce some necessary notation. Throughout the article, we use Ω to denote a bounded domain in R n , where n ≥ 2. For any x = (x 1 , . . . , x n ) ∈ Ω and r > 0,we write Ω r (x) = Ω ∩ B r (x), where B r (x) is the usual Euclidean ball of radius r centered at x. We also denote For a function f on Ω, we denote the average of f in Ω to be We use the notation For 1 ≤ q ≤ ∞, we define the space L q 0 (Ω) as the family of all functions u ∈ L q (Ω) satisfying (u) Ω = 0. We denote by W 1,q (Ω) the usual Sobolev space and W 1,q 0 (Ω) the closure of C ∞ 0 (Ω) in W 1,q (Ω). Let f , f α ∈ L q (Ω) n and g ∈ L q 0 (Ω). We say that (u, p) ∈ W 1,q 0 (Ω) n × L q 0 (Ω) is a weak solution of the problem and for any ϕ ∈ C ∞ 0 (Ω) n . Similarly, we say that (u, p) ∈ W 1,q 0 (Ω) n × L q 0 (Ω) is a weak solution of the problem if we have (2.1) and for any ϕ ∈ C ∞ 0 (Ω) n . Definition 2.1 (Green function). Let G(x, y) be an n × n matrix valued function and Π(x, y) be an n × 1 vector valued function on Ω × Ω. We say that a pair (G(x, y), Π(x, y)) is a Green function for the Stokes system (SP) if it satisfies the following properties: a) G(·, y) ∈ W 1,1 0 (Ω) n×n and G(·, y) ∈ W 1,2 (Ω \ B R (y)) n×n for all y ∈ Ω and R > 0. Moreover, Π(·, y) ∈ L 1 0 (Ω) n for all y ∈ Ω. b) For any y ∈ Ω, (G(·, y), Π(·, y)) satisfies div G(·, y) = 0 in Ω and L G(·, y) + DΠ(·, y) = δ y I in Ω in the sense that for any 1 ≤ k ≤ n and ϕ ∈ C ∞ 0 (Ω) n , we have c) If (u, p) ∈ W 1,2 0 (Ω) n ×L 2 0 (Ω) is the weak solution of (SP * ) with f , f α ∈ L ∞ (Ω) n and g ∈ L ∞ 0 (Ω), then we have Remark 2.2. The L 2 -solvability of the Stokes system with the Dirichlet boundary condition (see Section 3.1) and the part c) of the above definition give the uniqueness of a Green function. Indeed, if (G(x, y),Π(x, y)) is another Green function for (SP), then by the uniqueness of the solution, we have for any f ∈ C ∞ 0 (Ω) n and g ∈ C ∞ 0 (Ω). Therefore, we conclude that (G, Π) = (G,Π) a.e. in Ω × Ω.
2.1. Existence of the Green function. To construct the Green function, we impose the following conditions.
(A0). There exist positive constants R 1 and K 1 such that the following holds: for any x 0 ∈ ∂Ω and 0 < r ≤ R 1 , there is a coordinate system depending on x 0 and r such that in the new coordinate system, we have (A1). There exist constants µ ∈ (0, 1] and A 1 > 0 such that the following holds: if (u, p) ∈ W 1,2 (B R (x 0 )) n × L 2 (B R (x 0 )) satisfies where x 0 ∈ Ω and R ∈ (0, d x0 ], then we have where [u] C µ (B R/2 (x0)) denotes the usual Hölder seminorm. The statement is valid, provided that L is replaced by L * .
Then by the property c) of Definition 2.1 and the identity (2.5), we have the following representation for u: Also, the following estimates are easy consequences of the identity (2.5) and the estimates i) -v) in Theorem 2.3 for G * (·, x): In the theorem and the remark below, we show that if the coefficients have a vanishing mean oscillation (VMO), then the condition (A1) holds. If (u, p) ∈ W 1,2 (B R (x 0 )) n × L 2 (B R (x 0 )) satisfies (2.3) with x 0 ∈ Ω and 0 < R ≤ min{d x0 , 1}, then for any µ ∈ (0, 1), the estimate (2.4) holds with the constant A 1 depending only on n, λ, µ, and the VMO modulus of the coefficients. Remark 2.6. In the above theorem, the constant min{d x0 , 1} is interchangeable with min{d x0 , c} for any fixed c ∈ (0, ∞), possibly at the cost of increasing the constant A 1 . Setting c = diam Ω, the condition (A1) holds with the constant A 1 depending on n, λ, diam Ω, µ, and the VMO modulus ω ρ of coefficients.
The following corollary is immediate consequence of Theorem 2.3 and Remark 2.6.

2.2.
Global estimate of the Green function. We impose the following assumption to obtain the global pointwise estimate for the Green function.
From the global L q -estimates for the Stokes systems in Section 5, we obtain an example of the condition (A2) in the theorem below. The proof of the theorem follows a standard localization argument; see Section 4.4 for the details. Similar results for elliptic systems are given for the Dirichlet problem in [18] and for the Neumann problem in [5].
Let Ω be a domain in R n with diam(Ω) ≤ K 0 , where n ≥ 3. Assume the condition (A0) with a sufficiently small K 1 , depending only on n and λ. If the coefficients of L belong to the class of VMO, then the condition (A2) holds with the constant A 2 depending only on n, λ, K 0 , R 1 , and the VMO modulus of the coefficients.
By combining Theorems 2.8 and 2.9, we immediately obtain the following result. 3. Some auxiliary results.
3.1. L 2 -solvability. In this subsection, we consider the existence theorem for weak solutions of the Stokes system with measurable coefficients. For the solvability of the Stokes system, we impose the following condition.

(D).
Let Ω be a bounded domain in R n , where n ≥ 2. There exist a linear operator B : L 2 0 (Ω) → W 1,2 0 (Ω) n and a constant A > 0 such that div Bg = g in Ω and Bg W 1,2 0 (Ω) ≤ A g L 2 (Ω) . Remark 3.1. It is well known that if Ω is a Lipschitz domain with diam(Ω) ≤ K 0 , which satisfies the condition (A0), then for any 1 < q < ∞, there exists a bounded linear operator B q : where the constant C depends only on n, q, K 0 , K 1 , and R 1 ; see e.g., [1]. We point where C = C(n, q).

JONGKEUN CHOI AND KI-AHM LEE
Lemma 3.2. Assume the condition (D). Let For f ∈ L q (Ω) n , f α ∈ L 2 (Ω) n , and g ∈ L 2 0 (Ω), there exists a unique solution Moreover, we have In the case when
Proof. We mainly follow the argument given by Maz'ya-Rossmann [22,Theorem 5.2]. Also see [25,Theorem 3.1]. Let H(Ω) be the Hilbert space consisting of functions u ∈ W 1,2 0 (Ω) n such that div u = 0 and H ⊥ (Ω) be orthogonal complement of H(Ω) in W 1,2 0 (Ω) n . We also define P as the orthogonal projection from W 1,2 0 (Ω) n onto H ⊥ (Ω). Then, one can easily show that the operator B = P • B : Now, let f , f α ∈ L 2 (Ω) n and g ∈ L 2 0 (Ω). Then from the above argument, there exists a unique function w := Bg ∈ H ⊥ (Ω) such that (3.5) is satisfied. Also, by the Lax-Milgram theorem, one can find the function v ∈ H(Ω) that satisfies for all ϕ ∈ H(Ω). By setting ϕ = v in the above identity, and then, using Hölder's inequality and the Sobolev inequality, we have where q = 2 if n = 2 and q = 2n/(n + 2) if n ≥ 3. Therefore, the function u = v + w satisfies div u = g in Ω and the following identity: Moreover, we have To find p, we let is a bounded linear functional on L 2 (Ω). Therefore, there exists a function p 0 ∈ L 2 (Ω) so that and thus, p = p 0 − (p 0 ) Ω ∈ L 2 0 (Ω) also satisfies the above identity. Then by using the fact that B(L 2 0 (Ω)) = H ⊥ (Ω), we obtain for all ϕ ∈ H ⊥ (Ω). From (3.6) and (3.8), we find that (u, p) is the weak solution of the problem (3.2). Moreover, by setting ϕ = Bp in (3.8), we have and thus, we get (3.3) from (3.7). To establish (3.4), we observe that . By using the above inequality and (3.1), and following the same argument as above, one can easily show that the estimate (3.4) holds. The lemma is proved.

3.2.
Interior estimates. In this subsection we derive some interior estimates of u and p. We start with the following Caccioppoli type inequality that can be found, for instance, in [7,14].
where x 0 ∈ R n and R > 0. Then we have where C = C(n, λ).
by testing with φ in (3.9), we get where C 1 = C 1 (n, λ). From the above inequality, it remains us to show that Let 0 < ρ 1 < ρ 2 ≤ R and δ ∈ (0, 1). Let η be a smooth function on R d such that Then by applying η 2 u as a test function to and using the fact that div u = 0, we have and thus, by the ellipticity condition, Hölder's inequality, and Young's inequality, we obtain where C δ = C δ (n, λ, δ), and C 1 is the constant in (3.10). From this together with (3.10), it follows that Let us set where C = C(n, λ). By multiplying both sides of the above inequality by δ k and summing the terms with respect to k = 0, 1, . . ., we obtain By subtracting the last term of the right-hand side in the above inequality, we obtain the desired estimate (3.11). The lemma is proved.

15)
where C 2 = C 2 (n, µ, A 1 ). The statement is valid, provided that L is replaced by L * .
With the preparations in the previous section, we obtain the pointwise estimate of the averaged Green function G ε (·, y).

4.2.
Proof of Theorem 2.5. The proof is based on L q -estimates for Stokes systems with VMO coefficients. In this proof, we assume that x 0 ∈ Ω and 0 < R ≤ min{d x0 , 1}, and denote B r = B r (x 0 ) for r > 0.
where the coefficients of L belong to the class of VMO. Then we have where C depends on n, λ, q, and the VMO modulus of the coefficients.
Proof. Let τ = (ρ + r)/2 and η be a smooth function in R 2 such that

By Corollary 5.3 with scaling, we have
where C depends on n, λ, q, and the VMO modulus of the coefficients. Note that v L q (Br 1 ) ≤ C r 1 v L nq/(n+q) (Br 1 ) + C Dv L nq/(n+q) (Br 1 ) (4.29) for 0 < r 1 ≤ r. Combining the above two estimates we have . Using φ as a test function, we obtain Br) . From this together with (4.30), we get the desired estimate.

JONGKEUN CHOI AND KI-AHM LEE
Now we are ready to prove Theorem 2.5. Let (u, p) ∈ W 1,2 (B R ) n × L 2 (B R ) satisfy (2.3). Let q > n, 0 < r ≤ R, and ρ = r/4. Set where m is the smallest integer such that m ≥ n(1/2 − 1/q). Then by applying Lemma 4.5 iteratively, we see that (u, p) ∈ W 1,q (B ρ ) n × L q (B ρ ) and Using Hölder's inequality and Lemma 3.3, we have By the Sobolev inequality with scaling, we get where C depends on n, λ, q, and the VMO modulus of the coefficients. Since the above inequality holds for all x 0 ∈ Ω and 0 < r ≤ R ≤ min{d x0 , 1}, we conclude that This completes the proof of Theorem 2.5.
The theorem is proved.
For y ∈ Ω and r > 0, we denote B r = B r (y) and Ω r = Ω r (y).

By Corollary 5.3, we have
Using the fact that where C depends on n, λ, K 0 , R 1 , q, and the VMO modulus of the coefficients.

STOKES SYSTEMS 2011
We apply Caccioppoli's inequality (see, for instance, [17]) to the above estimate to get Step 3. We extend u to R n by setting u ≡ 0 on R n \ Ω. For y ∈ Ω and 0 < r < diam Ω, we obtain by (4.29) and (4.34) that Using this together with the Sobolev inequality, we have Since the above estimate holds for any y ∈ Ω and 0 < r < diam Ω, by using a standard argument (see, for instance, [13, pp. 80-82]), we derive This completes the proof of Theorem 2.9.

Remark 5.2.
We remark that γ-Reifenberg flat domains with a small constant γ > 0 satisfy the condition (D). Indeed, γ-Reifenberg flat domains with sufficiently small γ are John domains (and NTA-domains) that satisfy the condition (D). We refer to [2,1,19] for the details.
Since Lipschitz domains with a small Lipschitz constant are Refineberg flat, we obtain the following result from Theorem 5.1.
Proof. It suffices to prove the corollary with f = (f 1 , . . . , f n ) = 0. Indeed, by the solvability of the divergence equation in Lipschitz domains, there exist φ i ∈ W 1,q1 0 (Ω) n such that Due to Lemma 3.2, it is enough to consider the case q = 2. Case 1. q > 2. Let γ = γ(n, λ, q) and M = M (n, q) be constants in Theorem 5.1 and [11, Theorem 2.1], respectively. Set L = min{γ, M }. If K 1 ∈ (0, L], then by Theorem 5.1, the method of continuity, and the L q -solvability of the Stokes systems with simple coefficients (see [11,Theorem 2.1]), there exists a unique solution (u, p) ∈ W 1,q 0 (Ω) n × L q 0 (Ω) of the problem (5.1) with f = 0. Case 2. 1 < q < 2. We use the duality argument. Set q 0 = q q−1 , and let L = L(n, λ, q 0 ) and M = M (n, q) be constants from Case 1 and [11, Theorem 2.1], respectively. Assume that K 1 ≤ L and (u, p) ∈ W 1,q 0 (Ω) n × L q 0 (Ω) satisfies (5.1) where L * is the adjoint operator of L . Then we have which implies that where the constant C depends on n, λ, K 0 , R 1 , q, and the VMO modulus of the coefficients. Since h α was arbitrary, it follows that To estimate p, let w ∈ L q0 (Ω) and w 0 = w − (w) Ω . Then by Remark 3.1, there exists φ ∈ W 1,q0 (Ω) n such that By testing φ in (5.1), it is easy to see that This together with (5.4) yields Using the above L q -estimate, the method of continuity, and the L q -solvability of the Stokes systems with simple coefficients, there exists a unique solution (u, p) ∈ W 1,q 0 (Ω) n × L q 0 (Ω) of the problem (5.1) with f = 0.

Auxiliary results.
Lemma 5.4. Recall the notation (5.2). Suppose that the coefficients of L are constants. Let k be a constant.
then there exists a constant C = C(n, λ) such that then there exists a constant C = C(n, λ) such that Proof. The interior and boundary estimates for Stokes systems with variable coefficients were studied by Giaquinta [14]. The proof of the assertion (a) is the same as that of [14,Theorem 1.10,. See also the proof of [14,Theorem 2.8,p. 207] for the boundary estimate (5.6). We note that in [14], he gives the complete proofs for the Neumann problem and mentioned that the method works for other boundary value problem. Regarding the Dirichlet problem, we need to impose a normalization condition for p because (u, p + c) satisfies the same system for any constant c ∈ R. By this reason, the right-hand sides of the estimates (5.5) and (5.6) contain the L 2 -norm of p. For more detailed proof, one may refer to [7]. Their methods are general enough to allow the coefficients to be measurable in one direction and gives more precise information on the dependence of the constant C.
Next, let us set (u 2 , p 2 ) = (u, p) − (u 1 , p 1 ). Then, it is easily seen that (u 2 , p 2 ) = (0, 0) in Ω R \ B γ R and (u 2 , p 2 ) satisfies By Lemma 5.4, we get , and thus, from (5.18) and (5.9), we obtain (5.10). This completes the proof of the theorem. Now, we recall the maximal function theorem. Let For a function f on a set Ω ⊂ R n , we define its maximal function M(f ) by Then for f ∈ L q (Ω) with 1 < q ≤ ∞, we have where C = C(n, q). As is well known, the above inequality is due to the Hardy-Littlewood maximal function theorem. Hereafter, we use the notation A(s) = {x ∈ Ω : U (x) > s}, With Theorem 5.5 in hand, we get the following corollary.
Hereafter, we denote by U = {B Rx (x) : x ∈ A }, where A is the set of all points x ∈ A such that r x exists. Then by the Vitali lemma, we have a countable subcollection G such that (a) Q ∩ Q = ∅ for any Q, Q ∈ G satisfying Q = Q .
(c) |A| = |A | ≤ 5 n Q∈G |Q|. By the assumption (i) and (6.5), we see that Using this together with the assumption (ii) and Lemma 6.