VARIABLE LORENTZ ESTIMATE FOR STATIONARY STOKES SYSTEM WITH PARTIALLY BMO COEFFICIENTS

. We prove a global Calder´on-Zygmund type estimate in the framework of Lorentz spaces for a variable power of the gradients of weak solution pair ( u,P ) to the Dirichlet problem of stationary Stokes system. It is mainly assumed that the leading coeﬃcients are merely measurable in one spatial vari- able and have suﬃciently small bounded mean oscillation (BMO) seminorm in the other variables, the boundary of underlying domain is Reifenberg ﬂat, and the variable exponents p ( x ) satisfy the so-called log-H¨older continuity.


Introduction.
Let Ω ⊂ R n (n ≥ 2) be a bounded domain with a rough boundary specified later. We write the unknown velocity vectorial-value functions u = (u 1 , u 2 , · · · , u n ) : Ω −→ R n , and the pressure P : Ω → R. The main purpose of this present article is in minimizing regular requirements on the given datum to study a global Calderón-Zygmund type estimate in the framework of Lorentz spaces for a variable power of the gradients of weak solution pair (u, P ) to the Dirichlet problem of stationary Stokes system as follows:      D α (A αβ D β u) + ∇P = D α f α in Ω, div u = g in Ω, u = 0 on ∂Ω. (1) Throughout this paper, we use the Einstein summation convention on repeated indices. As usual, we suppose that each matrix-value entry of the coefficients A αβ = (A αβ ij ) n i,j : Ω → R n×n for every α, β = 1, 2, · · · , n satisfies the boundedness and the strong ellipticity, which means that there exists a constant 0 < Λ < 1 with with ξ α ∈ R n 2 for α = 1, 2, · · · , n. The weak solution of (1) is understood in the following usual sense, if for (u, P ) ∈ W 1,2 (Ω) n × L 2 (Ω) it holdŝ Ω A αβ D β u · D α ϕdx +ˆΩ P · div ϕdx =ˆΩ f α · D α ϕdx, ∀ϕ ∈ C ∞ 0 (Ω) n .
The solvability and optimal regularity for stationary Stokes system under minimal regular datum are always classical and important problems in the theory of partial differential equations and fluid dynamics. In recent years, there have been a great deal of literatures concerning the interior regularity of Stokes system (cf. [7]) and global regularity of generalized Stokes system in the Lipschitz domain (cf. [17,18,20,21]) or the rough domain with Reifenberg flat boundary (cf. [10,19,23]). Particularly, we would like to mention some recent advances concerning the generalized stationary Stokes problems (1) with discontinuous coefficients. Byun and So [10] recently considered the following generalized Stokes system: in Ω, u = 0 on ∂Ω, and they obtained the global weighted L q -estimate for the gradient of weak solution pair (u, P ) under some weak regular assumptions that the coefficients have small bounded mean oscillation (BMO) seminorm and the boundary of domain is Reifenberg flat, which implies the fact that F ∈ L q ω (Ω) n 2 ⇒ ∇u ∈ L q ω (Ω) n 2 , P ∈ L q ω (Ω). Further, Choi and Lee in [15] proved global a priori L q -estimate for stationary Stokes systems (1) just replacing D α f α by f + D α f α with BMO coefficients on the Reifenberg domain. On the other hand, Dong and Kim [18] also studied the same stationary Stokes systems with partially BMO coefficients on the bounded Lipschitz domains with small Lipschitz constant, and they derived Du L q (Ω) + P L q (Ω) ≤ c f L q 1 (Ω) + f α L q (Ω) + g L q (Ω) for q 1 ≥ nq q+n . Later, Choi, Dong and Kim in [14,19] extended it to the global weighted L q -estimates for the Dirichlet problem and the conormal derivative problem for the stationary Stokes system with partial regular coefficients, respectively, while Ω is a Reifenberg flat domain. Furthermore, Choi and Dong [13] showed that the solution (Du, P ) is bounded and its certain linear combinations are continuous if the coefficients are assumed to be merely measurable in one direction and have Dini mean oscillations in the other directions. Inspired by Choi, Dong and Kim's considerations above, this present article allows the coefficients A αβ are merely measurable in one direction and have a sufficiently small Bounded Mean Oscillation (BMO) seminorm in the other directions, such that our estimate is in the frame of Lorentz spaces for the variable power of the gradients of weak solution.
It is well-known fact that there have been a lot of research activities on the Calderón-Zygmund theory for various elliptic and parabolic problems with discontinuous coefficients. Except an earlier technique that used singular integral operators and their commutators, there were mainly three kinds of main different arguments to handle the Calderón-Zygmund theory concerning elliptic and parabolic problems with VMO or small BMO discontinuous coefficients. The first one is so-called geometrical approach originally traced from Byun and Wang's work in [9], which is used to attain global L p estimate based on the weak compactness, the boundedness of the Hardy-Littlewood maximal operators and the modified Vitali covering for distributional functions regarding the gradients of solutions. Here, the so-called modified Vitali covering actually refers to the argument as "crawling of ink spots" as in the early papers by Safonov and Krylov [27,31]. Indeed, this is also a development from Caffarelli and Peral's paper in [12] to obtain local W 1,p loc -estimates for solutions of p-Laplacian type elliptic problems with the nonlinearity satisfying the so-called Cordes-Nirenberg condition. Secondly, Kim and Krylov (cf. [25,26]) gave a unified approach of studying L p solvability for elliptic and parabolic problems in accordance with the Fefferman-Stein theorem on the sharp functions and the Hardy-Littlewood maximal function theorem for the spatial derivatives of solution. In this present paper, we have to highlight the third technique being called large-M-inequality principle originating from Acerbi and Mingione's work [2,3], which is directly applied to argue on certain Calderón-Zygmund type covering arguments instead of the classical maximal function operator and other harmonic analysis techniques such as the good-λ inequality. Furthermore, we would like to mention that Byun et al very recently have got numerous global Calderón-Zygmund type results to various nonlinear elliptic and parabolic problems over nonsmooth domains by combining large-M-inequality principle with their geometrical approach. Regarding our consideration, we particularly point out that Byun, Ok and Wang [8] first attained a global Calderón-Zygmund estimate with a variable exponent of the gradients of weak solution to the Dirichlet problem of linear elliptic system in divergence form with partially BMO coefficients and log-Hölder continuity p(x), which implies that Also, we mention that Tian and Zheng in [32] further generalized the above result to the global Calderón-Zygmund type estimate in Lorentz spaces for a variable power of the gradients of weak solution to the same problem on a rough domain with partial BMO coefficients.
Lorentz space is a two-parameter scale of the Lebesgue space obtained by refining it in the fashion of a second index, and there are a large body of literatures concerning Lorentz regularity for partial differential equations. For examples, Mengesha and Phuc [28] derived the weighted Lorentz estimate for the gradients of weak solution to quasilinear p-Laplacian type equations based on the geometric approach. Meanwhile, Baroni [5,6] obtained interior Lorentz estimate for the gradients of solutions to evolutionary p-Laplacian systems and obstacle parabolic p-Laplacian with the given obstacle function Dψ ∈ L(γ, q) locally in Ω T , respectively, by using the large-M-inequality principle, which yields the fact that F, Dψ ∈ L(γ, q) locally in Ω T ⇒ Du ∈ L(γ, q) locally in Ω T with γ > p and q ∈ (0, ∞]. Later, Zhang and Zhou [34] extended the above result in [28] to that for general elliptic equations of p(x)-Laplacian also using a geometrical argument, Adimurthil and Phuc [4] also proved the global Lorentz and Lorentz-Morrey estimates for quasilinear equations below the natural exponent. Very recently, Tian and Zheng [33] showed a global weighted Lorentz estimate to linear elliptic equations with lower order items under partially BMO coefficients in Reifenberg flat domain. Zhang and Zheng [35,36] also studied with Hessian Lorentz estimates for fully nonlinear parabolic and elliptic equations with small BMO nonlinearities, and weighted Hessian Lorentz estimates of strong solution for nondivergence linear elliptic equations with partially BMO coefficients, respectively. Now let us start with related basic notation being useful in the context. The Lorentz space L(t, q)(U ) for open subset U ⊂ R n with parameters 1 ≤ t < ∞ and 2882 SHUANG LIANG AND SHENZHOU ZHENG 0 < q < ∞, is the set of all measurable functions g : U → R requiring while the Lorentz space L(t, ∞) for 1 ≤ t < ∞ and q = ∞ is defined by the Marcinkiewicz space M t (U ) as usual, which is the set of all measurable functions g with The local variant of such spaces is defined as usual way. If t = q, then the Lorentz space L(t, t)(U ) is nothing but a classical Lebesgue space. Indeed, by Fubini's theorem it yields which implies L t (U ) = L(t, t)(U ), see also [5,6,28,35]. A usual assumption on the variable exponent p(·) is the so-called log-Hölder continuity, which ensures that the Hardy-Littlewood maximal operator is bounded within the framework of generalized Lebesgue space. To this end, we recall the definition that p(x) is log-Hölder continuous denoted it by p(x) ∈ LH(Ω), if there exist positive constants c 0 and δ such that for all x, y ∈ Ω with |x − y| < δ one has .
In the context, we assume that p(x) : Ω → R is a log-Hölder continuous function and there exist positive constants γ 1 and γ 2 such that Without loss of generality, let where ω : [0, ∞) → [0, ∞) is a modulus of continuity of p(x) such that ω is a nondecreasing continuous function with ω(0) = 0 and lim sup r→0 ω(r) log 1 r < ∞.
With the above assumptions in hand, it is clear that p(x) ∈ LH(Ω) yields that there exists a positive number A such that Before stating our main result, let us recall some basic concepts and related facts. We denote a type point by , and a typical boundary Ω r (y) = B r (y) ∩ Ω. For any f ∈ L 1 (U ) with a bounded measurable subset U ⊂ R n , we denote an average of f on U by Now we recall that Ω is a Reifenberg flat domain in the following sense.

VARIABLE LORENTZ ESTIMATE FOR STATIONARY STOKES SYSTEM
2883 Assumption 1.1. There exists R 0 ∈ (0, +∞) such that for each x 0 ∈ ∂Ω and each 0 < r ≤ R 0 , there exists a coordinate system depending only on x 0 and r, such that in the new coordinate system with the origin at x 0 , we have We are in a position to assume the regular assumptions on the nonlinearity A αβ (x) = A αβ (x 1 , x ) along with the rough boundary of Reifenberg domain Ω.
16 be a small constant to be specified later. We say that (A αβ , Ω) is (δ, R)-vanishing of codimension 1 if there exists R ∈ (0, R 0 ] such that the following conditions hold: (i) For x 0 ∈ Ω and 0 < r ≤ min{R, dist(x 0 , ∂Ω)}, there exists a coordinate system depending only on x 0 and r, whose variables are still denoted by x, such that in the new coordinate system we have (ii) For each x 0 ∈ ∂Ω and 0 < r < R, there exists a coordinate system depending only on x 0 and r, such that in the new coordinate system with the origin at x 0 such that (8) holds and The domain with (9) is called (δ, R)-Reifenberg flat domain, which is stronger than Assumption 1.1. Note that the stationary Stokes system has a scaling invariance property, which leads to that a small positive constant δ is still invariant under such a scaling so that (δ, R)-Reifenberg flat domain is also a non-tangentially accessible domain. We would remark that δ > 0 is determined by a set of parameters including the boundary flatness and R 0 in Assumption 1.1, see Remark 2.3 in [19] and Remark 1.5 in [11]. Notice that for a (δ, R)-Reifenberg flat domain we see that the boundary might be very rough between hyperplanes in a sufficiently small region going beyond the boundaries with C 1 -smooth or the Lipschitz category with a small Lipschitz constant. Moreover, it is obvious fact that this is A-type domain, which ensures the following measure density condition (cf. [8]): Finally, let us summarize our main result of this paper as follows.
This article is to focus on a global Calderón-Zygmund type estimate in the framework of Lorentz spaces for a variable power of the gradients of weak solutions to the Dirichlet problem of stationary Stokes system (1) with partially regular coefficients on Reifenberg domains. Our problem and its argument are inspired to a great extent by work of Acerbi and Mingione [2,3] and Baroni [5,6], and recent work from Byun and So [10]. The point is to make use of the mixed argument of large-M-inequality principle and geometric approach to the Calderón-Zygmund type covering on the super-level set E (λ, Ω R ). Indeed, our main key ingredient is based on our using Calderón-Zygmund type covering, approximate estimate and an iteration argument to obtain an estimate of the measure of the super-level set for the variable power of the gradient of a solution.
The remainder of the paper is organized as follows. In Section 2, we introduce some useful lemmas. In Section 3, we focus on proving our main theorem.
2. Technical tools. In this section we present some useful lemmas, which will play essential roles in proving our main conclusion. From then on, we denote by C i (n, Λ, · · · ) or c i (n, Λ, · · · ) for i = 1, 2, · · · , the universal constant depending only on prescribed quantities and possibly varying from line to line. First, let us do impose nothing regularity on the coefficients A α, β (x), but a relaxed geometric requirement on the domain Ω.
Nečas had proved L p -version of the Babuška-Aziz inequality for 1 < p < ∞ on Lipschitz domains. Acosta, Durán and Muschietti [1] extended solvability of the divergence equation on John domains Ω ⊂ R n with n ≥ 2 via a constructive approach, see also [16]. If Ω is a bounded Reifenberg flat domain, then the domain is also a John domain. Therefore, we know that the domain Ω satisfies Definition 2.1 with a constant C 1 depending only on n, R 0 and diam(Ω). More precisely, for a bounded domain we have Reifenberg flat domains ⊂ NTA-domains ⊂ uniform domains ⊂ John domains, where NTA-domains are briefly written as the non-tangentially accessible domains, see [19,24].
Let us recall the existence and energy estimate of the weak solution pair to stationary Stokes system (1) defined in a more general domain with the Babuška-Aziz inequality, see Lemma 3.4 in [19].
Proceeding as in the same way as the proof of Lemma 3.8 in [19], we get (13).
iii) If |g| α ∈ L(t, q)(U ) for some 0 < α < ∞, then g ∈ L(αt, αq)(U ) with the estimate The following two lemmas will play important roles in our main proof, which are the variants of classical Hardy's inequality and a reverse Hölder inequality, respectively, see Lemma 3.4 and 3.5 in [5].
Then for any α ≥ 1 and r > 0, it holds that
Finally,let us recall the following iteration argument in the main proof, see [22] or Lemma 4.1 in [30].
where B 0 , L ≥ 0 and β > 0. Then 3. Proof of Theorem 1.3. This section is mainly devoted to proving Theorem 1.3 via a combination of the so-called large-M-inequality principle in [3] and the geometric argument in [8]. For this, we here part it in six steps. In step 1, for given λ 0 in (26), we show the Calderón-Zygmund type covering on the super-level set E (λ, Ω R ), and establish the decay estimates of Ω ry (y). In step 2, we give various comparison estimates with the reference problems so as to get the comparison estimate with the limiting one. In step 3, we employ the so-called " crawling of ink spots" approach to show an essential estimate for the super-level set of the gradients with a variable power of the gradients. In steps 4 and 6, we attain our conclusions in the cases of q < ∞ and q = ∞, respectively, under a priori assumption (|∇u| + |P |) p(x) L(t,q)(Ω 2R ) < ∞ which will be proved in step 5.
Proof. Note that Ω is bounded (δ, R 0 )-Reifenberg domain, then it holds the Babuška-Aziz inequality with C 1 depending only on n, |Ω|, also see Remark 3.3 in [19]. In the following we will use Lemma 2.2, Lemma 2.3 and Lemma 2.4 uniformly with For the weak solution pair (u, P ) of original problem and the nonhomogeneous terms f α , g, by a scaling argument we writẽ Then, by the assumption |f α | + |g| p(x) ∈ L(t, q)(Ω) we have Hereafter, for the sake of simplicity, we still use u, P, f α and g replacingũ,P ,f α andg in the following.
Step 1. In this step, we make use of the Calderón-Zygmund type covering on the super-level set E (λ, Ω R ) to show a decay estimates of Ω ry (y). Let u be the weak solution of (1). For any x 0 ∈ Ω and Ω R = Ω R (x 0 ), we define the quantity where δ > 0 and η > 1 will be specified later. We now introduce the super-level set n . For y ∈ E (λ, Ω R ) and radii 0 < r ≤ R, we let CZ(Ω r (y)) := Note that R 400 ≤ r ≤ R. By simply enlarging the domain of integration we get which means that while R 400 ≤ r ≤ R one has CZ(Ω r (y)) < λ.
On the other hand, by Lebesgue's differentiation theorem we get that for 0 < r 1 CZ(Ω r (y)) > λ.
Therefore, by absolute continuity of the integral with respect to the domain we can pick the maximal radius r y such that CZ(Ω ry (y)) =
From (28) we conclude the following alternatives: First, we suppose that the first case of (30) is valid and we split it as follows: where σ 1 > 0 is determined later.
For the case of the second estimate in (30), by taking ζ = δ 4 , Fubini's theorem and a split of the integral we get Let δ = 4ζ, we derive Now we put (33) and (34) together, and get Ω ry (y) ≤c Ω ry (y) ∩ E( λ 4 , Ω 2R ) Step 2. This step is devoted to various comparison estimates with the reference problems and the limiting one. Taking into account (29), we have Therefore, it suffices to show that for a constant c 3 ≥ 1. We first claim that Ω400r y (y) with c ≥ 1 a universal constant. In fact, since |f α | + |g| ∈ L(t, q)(Ω) for t > 1 and 0 < q ≤ ∞, it deduces that where we used (25). By L 2 -estimate in Lemma 2.2, it leads to that Ω |Du| 2 + |P | 2 dx ≤ c 1 + |Ω| .
That is, Using "crawling of ink spots" argument again and (35), we conclude that Step 4. This step is devoted to proving our main conclusion under the claim that |∇u| + |P | p(x) L(t,q)(Ω 2R ) < ∞ for 0 < q < ∞. Since t > 1, we multiply the inequality (43) by tp −