# \(L_q\)-estimates for stationary Stokes system with coefficients measurable in one direction

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## Abstract

We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori \(\dot{W}^1_q\)-estimates for any \(q\in [2,\infty )\) when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a \(W^1_q\)-estimate and prove the solvability for any \(q\in (1,\infty )\) when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal directions in each small ball, where the direction is allowed to depend on the ball.

## Keywords

Stokes systems Boundary value problem Measurable coefficients## Mathematics Subject Classification

35R05 76N10 35B65## 1 Introduction

*q*restricted, using the layer potential method and Rellich identities. For this line of research, see also [6, 19, 30, 31, 32] and references therein, some of which obtain estimates in Besov spaces.

In this paper, we allow coefficients \(A^{\alpha \beta }\) to be merely measurable in one direction. In particular, they may have jump discontinuities, so that the system can be used to model, for example, the motion of two fluids with interfacial boundaries. The system (1.2) is considered in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori \(\dot{W}^1_q\)-estimates for any \(q\in [2,\infty )\) in the case that the coefficients are merely measurable functions in one fixed direction (see Theorem 2.1). For the system on bounded Lipschitz domains with a small Lipschitz constant, we prove a \(W^1_q\)-estimate and solvability for (1.2) with any \(q\in (1,\infty )\), when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal directions in each small ball, with the direction depending on the ball (see Theorem 2.6). These results extend the aforementioned results from [18] for the classical Stokes system (1.1), and the recent study [9] for (1.2) with coefficients having small mean oscillations in all directions. We note that the class of coefficients considered in this paper was first introduced by Kim and Krylov [23] and Krylov [25], where they established the \(W^2_p\)-estimate for non-divergence form second-order elliptic equations in the whole space. Subsequently, such coefficients were also treated in [13, 14] for second- and higher-order elliptic and parabolic systems in regular and irregular domains.

*Du*. To this end, instead of considering

*Du*itself, we exploit an idea given in [14] to estimate certain linear combinations of

*Du*and

*p*:

*u*in the \(x_i\) direction, \(i=2,\ldots ,d\), where \(x = (x_1, x') = (x_1, x_2, \ldots ,x_d) \in \mathbb {R}^d\). For the Stokes system, the presence of the pressure term

*p*gives an added difficulty, because in the usual \(L_2\)-estimate, instead of

*p*one can only bound \(p-(p)\) by

*Du*, instead of

*p*. See, for instance, Lemma 3.5. Nevertheless, in Lemma 4.2 we show that for the homogeneous Stokes system and any integer \(k\ge 1\), the \(L_2\)-norm of \(D_{x'}^k p\) in a smaller ball can be controlled by that of

*Du*in a larger ball. Finally, in order to deal with the system (1.2) in a Lipschitz domain, we apply a version of the Fefferman–Stein sharp function theorem for spaces of homogeneous type, which was recently proved in [12] (cf. Lemma 7.3). Furthermore, we employ a delicate cut-off argument, together with Hardy’s inequality, which was first used in [13] and also in the recent paper [9].

In a subsequent paper, we will study *weighted* \(W^1_q\)-estimates and the solvability of the Stokes system (1.2) in more general Reifenberg flat domains, with the same class of coefficients. We note that an *a priori* \(W^1_q\)-estimate in Reifenberg flat domains was obtained in [9], under the condition that the coefficients have sufficiently small mean oscillations in all directions.

The remainder of this paper is organized as follows. We state the main theorems in the following section. Section 3 contains some auxiliary results, including \(L_2\)-estimates and Caccioppoli type inequalities. In Sect. 4, we prove interior and boundary \(L_\infty \) and Hölder estimates for derivatives of solutions, while in Sect. 5 we establish the interior and boundary mean oscillation estimates for the system in the whole space and in a half space. Section 6 is devoted to the proof of Theorem 2.1. Finally, we consider the system (1.2) in a Lipschitz domain with a small Lipschitz constant in Sect. 7.

*r*in \(\mathbb {R}^d\) centered at \(x_0\in \mathbb {R}^d\), and let \(B^+_r(x_0)\) be the half ball

*f*, we define its average on \(\Omega \) byWe shall use the following function spaces:

## 2 Main results

### Theorem 2.1

### Remark 2.2

In Theorem 2.1 we only consider the case that \(q \in [2,\infty )\) to simplify the exposition and to present our approach in the most transparent way. Indeed, if \(q = 2\), the theorem holds even with measurable \(A^{\alpha \beta }(x)\). See Theorem 3.4. Thus, in the proof of Theorem 2.1 we focus on the case \(q \in (2,\infty )\), the proof of which well illustrates, in the simplest setting, our arguments based on mean oscillation estimates together with the sharp function and the maximal function theorems. One can prove the other case, with \(q \in (1,2)\), by using Theorem 2.6 below. This will be discussed in a more general setting with weights in a forthcoming paper [15].

Next, when the Stokes system is defined in a bounded Lipschitz \(\Omega \) with a small Lipschitz constant, we show that the system is uniquely solvable in \(L_q(\Omega )\) spaces. In this case, we allow coefficients not only to be measurable locally in one direction (near the boundary the direction is almost perpendicular to the boundary of the domain), but also to have small mean oscillations in the other directions. To present this result, we require the following assumptions.

### Assumption 2.3

*r*such that in the new coordinate system we have

### Assumption 2.4

- (1)For \(x_0 \in \Omega \) and \(0 < r \le \min \{R_1, {\text {dist}}(x_0, \partial \Omega )\}\), there is a coordinate system depending on \(x_0\) and
*r*such that in this new coordinate system we have that(2.3) - (2)For any \(x_0 \in \partial \Omega \) and \(0 < r \le R_1\), there is a coordinate system depending on \(x_0\) and
*r*such that in the new coordinate system we have that (2.3) holds, andwhere \(\phi : \mathbb {R}^{d-1} \rightarrow \mathbb {R}\) is a Lipschitz function with$$\begin{aligned} \Omega \cap B_r(x_0) = \{x \in B_r(x_0)\, :\, x_1 >\phi (x')\}, \end{aligned}$$$$\begin{aligned} \sup _{\begin{array}{c} x', \, y' \in B_r'(x_0')\\ x' \ne y' \end{array}}\frac{|\phi (y')-\phi (x')|}{|y'-x'|}\le \rho . \end{aligned}$$

### Remark 2.5

Clearly, Assumption 2.4 (2) is stronger than Assumption 2.3. However, we state these two assumptions separately for the following reason. As seen in Theorem 2.6, we specify the class of bounded Lipschitz domains for which the results of the theorem hold in terms of the flatness parameter \(\rho \). Thus, having two separate assumptions means that in Theorem 2.6 we specify a subclass of the class of domains satisfying Assumption 2.3. The necessity of such a hierarchy of classes of domains is that when determining the size of \(\rho \) in Theorem 2.6, we need some information about domains and their boundaries. In particular, the maximal function and sharp function theorems on bounded domains we use in this paper require such information. Thus, without Assumption 2.3, the size of \(\rho \) is to be determined by a set of parameters including \(R_1\). In this case, i.e., when \(\rho \) is given by \(R_1\), even a smooth domain \(\Omega \) may not satisfy Assumption 2.4 (2) if \(R_1\) is too large for the boundary to have \(\rho \) flatness on \(\Omega \cap B_{R_1}(x_0)\). With two assumptions as above, every smooth domain satisfies Assumption 2.4 (2) for any \(\rho \) by choosing a sufficiently small \(R_1\).

### Theorem 2.6

*d*, \(\delta \), \(R_0\), \(R_1\),

*K*,

*q*, and \(q_1\). Moreover, for \(f \in L_{q_1}(\Omega )\), \(f_\alpha , g \in L_q(\Omega )\) with \((g)_\Omega = 0\), there exists a unique \((u,p) \in W_q^1(\Omega )^d \times L_q(\Omega )\) satisfying \((p)_\Omega = 0\) and (2.4).

## 3 Auxiliary results

In this section, we assume that the coefficients \(A^{\alpha \beta }\) are measurable functions of \(x \in \mathbb {R}^d\). That is, no regularity assumptions are imposed on \(A^{\alpha \beta }\).

We impose the following assumption on a bounded domain \(\Omega \subset \mathbb {R}^d\) in Lemma 3.3 below.

### Assumption 3.1

*d*and \(\Omega \) such that

### Remark 3.2

If \(\Omega = B_R\) or \(\Omega = B_R^+\), it follows from a scaling argument that the constant \(K_1\) depends only on the dimension *d*. If \(\Omega \) is a bounded Lipschitz domain satisfying Assumption 2.3, then Assumption 3.1 is satisfied with \(K_1\) depending only on *d*, \(R_0\), and \({\text {diam}} \Omega \). If 1 / 16 in Assumption 2.3 is replaced by \(\rho \) and \(\rho \in [0,\rho _0]\), the constant \(K_1\) can be chosen so that it depends only on *d*, \(R_0\), \({\text {diam}}\Omega \), and \(\rho _0\). See, for instance, [5].

### Lemma 3.3

### Proof

See, for instance, [9, Lemma 3.1]. \(\square \)

As far as a priori estimates are concerned, one can have \(\Omega =\mathbb {R}^d\) or \(\Omega = \mathbb {R}^d_+\) in Lemma 3.3 if \(f \equiv 0\). In this case we do not necessarily need that the integral of *p* over \(\Omega \) is zero. For completeness and later reference, we state and prove this result in the theorem below.

### Theorem 3.4

### Proof

*u*as a test function to obtain that

In the lemmas below, we do not necessarily have that \((p)_{B_R} = 0\) or \((p)_{B_R^+} = 0\) unless specified. We note that by now these lemmas are fairly standard results, and we present them here for the sake of completeness. See, for instance, [22] and [9], and also [20] under slightly different conditions on the coefficients.

### Lemma 3.5

### Proof

### Lemma 3.6

- (1)If \((u,p) \in W_2^1(B_R)^d \times L_2(B_R)\) satisfiesthen for any \(\varepsilon > 0\), we have that$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {L}u + \nabla p = 0 &{}\quad \text {in}\,\,B_R,\\ {\text {div}} u = 0 &{}\quad \text {in}\,\,B_R, \end{array} \right. \end{aligned}$$(3.6)where \(N = N(d,\delta ,\varepsilon )\).$$\begin{aligned} \int _{B_r} |Du|^2 \, dx \le N (R-r)^{-2} \int _{B_R}|u|^2 \, dx + \varepsilon \int _{B_R} |Du|^2 \, dx, \end{aligned}$$(3.7)
- (2)If \((u,p) \in W_2^1(B_R^+)^d \times L_2(B_R^+)\) satisfiesthen for any \(\varepsilon >0\), we have that (3.7) holds with \(B_r^+\) and \(B_R^+\) replacing \(B_r\) and \(B_R\), respectively, where \(N = N(d,\delta , \varepsilon )\).$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {L}u + \nabla p = 0 &{}\quad \text {in}\,\,B_R^+,\\ {\text {div}} u = 0 &{}\quad \text {in}\,\,B_R^+,\\ u=0 &{}\quad \text {on}\,\, B_R \cap \partial \mathbb {R}^d_+, \end{array} \right. \end{aligned}$$(3.8)

### Proof

We only prove the second assertion of the lemma, because the first is the same with obvious modifications.

### Lemma 3.7

- (1)If \((u,p) \in W_2^1(B_R)^d \times L_2(B_R)\) satisfies (3.6), then we have thatwhere \(N=N(d,\delta )\).$$\begin{aligned} \int _{B_r} |Du|^2 \, dx \le N (R-r)^{-2} \int _{B_R} |u|^2 \, dx, \end{aligned}$$(3.10)
- (2)
If \((u,p) \in W_2^1(B_R^+)^d \times L_2(B_R^+)\) satisfies (3.8), then we have that (3.10) holds, with \(B_r^+\) and \(B_R^+\) replacing \(B_r\) and \(B_R\), respectively.

### Proof

## 4 \(L_\infty \) and Hölder estimates

*Du*and

*p*, which are crucial for proving our main results. Recall the operator \(\mathcal {L}_0\) given in (2.1), where the coefficients are functions of \(x_1\) only. In this case, if a sufficiently smooth (

*u*,

*p*) satisfies \(\mathcal {L}_0 u + \nabla p = 0\) in \(\Omega \subset \mathbb {R}^d\), we see that

### Lemma 4.1

- (1)If \((u,p) \in W_2^1(B_R)^d \times L_2(B_R)\) satisfiesthen \(DD_{x'} u \in L_2(B_r)\), and$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {L}_0 u + \nabla p = 0 &{}\quad \text {in}\,\,B_R, \\ {\text {div}} u = \ell &{}\quad \text {in}\,\,B_R, \end{array} \right. \end{aligned}$$(4.3)where \(N=N(d,\delta )\).$$\begin{aligned} \int _{B_r} |DD_{x'}u|^2 \, dx \le N(R-r)^{-2} \int _{B_R} |Du|^2 \, dx, \end{aligned}$$(4.4)
- (2)If \((u,p) \in W_2^1(B_R^+)^d \times L_2(B_R^+)\) satisfiesthen \(DD_{x'}u \in L_2(B_r^+)\), \(D_{x'} u = 0\) on \(B_r \cap \partial \mathbb {R}^d_+\), and (4.4) is satisfied with \(B_r^+\) and \(B_R^+\) replacing \(B_r\) and \(B_R\), respectively.$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {L}_0 u + \nabla p = 0 &{} \quad \text {in}\,\,B_R^+,\\ {\text {div}} u = \ell &{}\quad \text {in}\,\,B_R^+, \\ u = 0 &{}\quad \text {on}\,\, B_R \cap \partial \mathbb {R}^d_+, \end{array} \right. \end{aligned}$$(4.5)

### Proof

*f*with respect to \(x_j\), i.e.,

To estimate *U*, we also need to bound \(D_{x'} p \in L_2(B_r)\), as in the following key lemma.

### Lemma 4.2

- (1)If \((u,p) \in W_2^1(B_R)^d \times L_2(B_R)\) satisfies (4.3) in \(B_R\), then \(D_{x'}p \in L_2(B_r)\) andwhere \(N = N(d,\delta )>0\).$$\begin{aligned} \int _{B_r}|D_{x'}p|^2\,dx\le N(R-r)^{-2} \int _{B_R}|Du|^2\,dx, \end{aligned}$$(4.7)
- (2)
If \((u,p) \in W_2^1(B_R^+)^d \times L_2(B_R^+)\) satisfies (4.5) in \(B_R^+\), then \(D_{x'}p \in L_2(B_r^+)\), and (4.7) is satisfied with \(B_r^+\) and \(B_R^+\) replacing \(B_r\) and \(B_R\), respectively.

### Proof

*u*defined by

### Lemma 4.3

### Proof

See the proof of Lemma 4.4 below, with obvious modifications. \(\square \)

### Lemma 4.4

### Proof

*U*and the sup-norm estimate of \(U_i\), \(i = 2, \ldots , d\). Set

*U*and \(\tilde{U}\), we have that

## 5 Mean oscillation estimates

In this section, we prove our mean oscillation estimates using the Hölder estimates developed in Sect. 4 and the \(L_2\)-estimates of the Stokes system given in Lemma 3.3. Throughout this section, we consider the operator \(\mathcal {L}_0\), i.e., the coefficients \(A^{\alpha \beta }\) are measurable functions of \(x_1\) only.

### Lemma 5.1

### Proof

This lemma follows as a consequence of Lemmas 3.3 and 4.3. See Case 2 in the proof of Lemma 5.2 below. \(\square \)

### Lemma 5.2

### Proof

Denote the first coordinate of \(x_0\) by \({x_0}_1\). We consider the following two cases.

**Case 1**\({x_0}_1 \ge \kappa r/8\). In this case, we have that

**Case 2**\({x_0}_1 < \kappa r/8\). Set \(y_0 = (0,x_0')\). Then, we have that

*V*is defined in exactly the same way as

*U*in (4.2) with

*v*in place of

*u*. Then, it follows from the triangle inequality that

We can similarly obtain the desired estimate for *U*. Thus, the lemma is proved. \(\square \)

## 6 Proof of Theorem 2.1

*x*, where \(n \in \mathbb {Z}\). The maximal function of

*f*in \(\mathbb {R}^d\) or \(\mathbb {R}^+\) is defined by

*x*with \(r > 0\), where \(x_0\in \bar{\Omega }\).

### Proof of Theorem 2.1

Because Theorem 3.4 covers the case with \(q=2\), we assume that \(q \in (2,\infty )\). We prove the case when \(\Omega = \mathbb {R}^d_+\). The other case is simpler.

*x*. From this and (6.2), we have that

*p*. To obtain \(L_q\)-estimates of such terms, we first note the relation

*d*, \(\delta \), and

*q*, such that \(N \kappa ^{-1/2} \le 1/2\), we arrive at

*p*is bounded by the right-hand side of (2.2). By this and (6.6), we can conclude that the estimate (2.2) holds, and the theorem is proved. \(\square \)

## 7 Proof of Theorem 2.6

### Lemma 7.1

### Proof

We mainly follow the proof of Proposition 7.10 in [13], where \(\mathbb {R}^d_+\) instead of \(\Omega \) is considered. Let \(\tilde{x}\in \partial \Omega \) be such that \(|x_0 -\tilde{x}|=\text {dist}(x_0,\partial \Omega )\). As in the proof of Lemma 5.2, we consider two cases.

**Case 1**\(|x_0 -\tilde{x}|\ge \kappa r/16\). In this case, we have that

*U*, we see that (7.2) is satisfied.

**Case 2**\(|x_0 -\tilde{x}|< \kappa r/16\). Without loss of generality, one may assume that \(\tilde{x}\) is the origin. Note that

- (1)
For any \(n\in \mathbb {Z}\), \(\mu (\Omega {\setminus } \bigcup _{\alpha }Q_\alpha ^n)=0\);

- (2)
For each

*n*and \(\alpha \in I_n\), there exists a unique \(\beta \in I_{n-1}\) such that \(Q_\alpha ^n\subset Q_\beta ^{n-1}\); - (3)
For each

*n*and \(\alpha \in I_n\), \(\text {diam}(Q_\alpha ^n)\le N_0\delta _0^n\); - (4)
Each \(Q_\alpha ^n\) contains some ball \(\Omega _{\varepsilon _0\delta _0^n}(z_\alpha ^n)\);

*d*, \(R_0\), and

*K*.

### Lemma 7.2

*d*,

*q*, \(R_0\), and

*K*.

### Proof

Since \(\Omega \) is a space of homogeneous type, the lemma follows from the Hardy–Littlewood maximal function theorem for spaces of homogeneous type. See, for instance, [4]. Also see [12, Theorem 2.2]. \(\square \)

### Lemma 7.3

*Q*such that

*d*,

*q*, \(R_0\), and

*K*.

### Proof

This lemma is a special case of Theorem 2.4 of [12], in which \(A_q\) weights are considered. When there is no weight as in the lemma, it is easily seen that \(\beta \) in that theorem is equal to 1. \(\square \)

We are now ready to present the proof of Theorem 2.6.

### Proof of Theorem 2.6

**Case 1**\(q>2\). We take \(\mu \in (1,\infty )\), depending only on

*q*, such that \(2\mu <q\), and we let \(\kappa \ge 32\) be a constant to be specified. By the properties (3) and (4) described above, for each

*Q*in the partitions there exist \(r \in (0,\infty )\) and \(x_0 \in \bar{\Omega }\) such that

*N*depends only on

*d*, \(R_0\), and

*K*. See Remark 7.3 in [12]. In order to apply Lemma 7.3, we take \(F=|Du|\), \(H=N|Du|\), where \(N = N(d,\delta ,q) \ge 1\) from (7.2), and

*d*, \(\delta \), \(R_0\),

*K*, and

*q*. Otherwise, i.e., if \(r>R_1/\kappa \) we take \(F^Q=|Du|\). Then, by (7.19) we havewhere \(N = N(d, R_0, K)\). Since \(|\Omega _{R_1/\kappa }(x_0)|^{-1}\le NR_1^{-d}\kappa ^{d}\), we still get that (7.17) holds with \(N_0\) depending only on

*d*, \(R_0\), and

*K*. Therefore, the conditions in Lemma 7.3 are satisfied. From (7.18), we obtain that

*d*, \(\delta \), \(R_0\),

*K*, and

*q*(but independent of \(R_1\)), so that

*p*. For any \(\eta \in L_{q'}(\Omega )\) with \(q'=q/(q-1)\), it follows from the solvability of the divergence equation in Lipschitz domains (cf. [5]) that there exists \(\psi \in \mathring{W}_{q'}^1(\Omega )^d\) such that

*d*, \(R_0\),

*K*, and \(q'\). We test the first equation of (2.4) by \(\psi \), and using the fact that \((p)_\Omega = 0\) we obtain

**Case 2**\(q\in (1,2)\). We employ a duality argument. Let \(q'=q/(q-1)\in (2,\infty )\) and \((\gamma , \rho ) = (\gamma , \rho )(d,\delta ,R_0,K,q')\) from Case 1. Then, for any \(\eta =(\eta _{\alpha })\), where \(\eta _\alpha \in L_{q'}(\Omega )^d\) for \(\alpha =1,\ldots ,d\), there exists a unique solution \((v,\pi ) \in W_{q'}^{1}(\Omega )^d \times L_{q'}(\Omega )\) with \((\pi )_\Omega = 0\) satisfying

*u*, to obtain

*p*is the same as in Case 1. Thus, the theorem is proved. \(\square \)

## Notes

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