A leading term for the velocity of stationary viscous incompressible flow around a rigid body performing a rotation and a translation

We consider the Navier-Stokes system with Oseen and rotational terms describing the stationary flow of a viscous incompressible fluid around a rigid body moving at a constant velocity and rotating at a constant angular velocity. In a previous paper, we proved a representation formula for Leray solutions of this system. Here the representation formula is used as starting point for splitting the velocity into a leading term and a remainder, and for establishing pointwise decay estimates of the remainder and its gradient.


1.
Introduction. Consider a rigid body moving with constant velocity and rotating with constant angular velocity in a viscous incompressible fluid. Suppose the flow around the body is steady ("developed"). Then the usual mathematical model of this flow, with respect to a reference frame in which the body is at rest, is given by the Navier-Stokes system with an Oseen term and rotational terms. Written in normalized form (see [41] for details), this system takes the form and is supplemented by homogeneous Dirichlet boundary conditions at infinity, |u(x)| → 0 for |x| → ∞.
In this model, the open bounded set D ⊂ R 3 corresponds to the rigid body, the unknown functions u and π describe respectively the velocity and the pressure field of the flow, whereas the given function f represents a prescribed volume force acting on the fluid. The parameter τ ∈ (0, ∞) is the Reynolds number, and ∈ R\{0} the Taylor number. We assume the Reynolds number and the Taylor number to be fixed throughout the paper. The symbol e 1 stands for the unit vector (1, 0, 0). The constant velocity of the body, as seen from an observer at rest, is given by −τ e 1 , and its constant angular velocity by e 1 . In particular, the direction of translation and the axis of rotation of the rigid body are parallel, as should be expected in the case of a steady-state flow around that body. We are interested in Leray solutions to (1), (2), that is, solutions (u, π) such that u ∈ L 6 (D c ) 3 , ∇u ∈ L 2 (D c ) 9 and π ∈ L 2 loc (D c ). This type of solution is known to exist even for large data, under suitable regularity assumptions on ∂D and f ([36, Theorem XI.3.1]). Note that the conditions u ∈ L 6 (D c ) 3 , ∇u ∈ L 2 (D c ) 9 mean in particular that (2) holds in a weak sense ([36, Theorem II.6.1]). Another property of Leray solutions that is well known by now concerns their asymptotics far from D. In fact, independently of any boundary condition on ∂D, the velocity part u of such a solution satifies the relation with α ∈ N 3 0 with |α| := α 1 + α 2 + α 3 ≤ 1, if |f (x)| tends to zero sufficiently fast for large values of |x| ( [37], [10]). Note that in the case α = 0, the relation in (3) describes the decay of u, whereas the asymptotic behavior of ∇u is specified by (3) if α ∈ N 3 0 with |α| = 1. The term s τ (x) is defined as s τ (x) := 1 + τ (|x| − x 1 ) (x ∈ R 3 ).
Since the vector −τ e 1 corresponds to the velocity of the body as seen from an observer at rest, the factor s τ (x) −1−|α|/2 in (3) should be considered as a mathematical manifestation of the wake extending behind a body moving in a viscous incompressible fluid. In view of (3), it is natural to look for a vector-valued function L ("leading term") with the property that L(x) decays with exactly the rate given by the right-hand side of (3) with α = 0, whereas the term (u−L)(x) ("remainder") decays pointwise at a rate which is higher than the one in (3) with α = 0. In addition, an analogous statement should hold for ∂ α L(x) if |α| = 1. Identifying such a function L is highly interesting from a physical point of view because it gives a good idea of the structure ("profile") of the flow as observed at some distance from the rigid body. This interpretation of L explains why a leading term should be expressed, if possible, as an explicit combination of elementary functions. Actually, beyond the physical meaning of such a term, it might even be argued that in any case, the notion of leading term refers to functions that are given in a more or less explicit way. A leading term is relevant also in some mathematical applications. For example, numerical computation of exterior flows often involve bounded computational domains, with an artificial boundary condition on the outer boundary of such domains. The choice of such boundary conditions, and related error estimates, are determined by the asymptotic structure of the flow field, and thus by a leading term; see [2,3,27].
For some types of flows, a leading term with all relevant features could be exhibited. This is true if = 0 and the Oseen term in (1) is dropped (case of a body which neither rotates nor translates, or which is rotationally symmetric and rotates but does not translate). In this situation, Korolev andŠverák [51] could show that L is given by the so-called Landau solution of the stationary Navier-Stokes system under the assumption that the velocity is small in a suitable sense. In a series of papers due to Farwig, Hishida [20], [21], [22] and Farwig, Galdi, Kyed [17], a satisfactory theory could also be developed for the case that the Oseen term τ ∂ 1 u is not present in (1) but the parameter does not vanish (flow around a body that rotates but does not translate; "purely rotational case"). In this situation, if is small, it turned out that a leading term is again given by the Landau solution. On the other hand, if the Oseen term is present but vanishes (case of a body that translates but does not rotate; "purely translational case"), it was shown first for solutions that are "strong" in a suitable sense (see [28], [29]), and then for Leray solution (see [4], [36] and the references in [36]) that a leading term may be given by an appropriate linear combination of the columns of the (matrix-valued) velocity part of a fundamental solution to the stationary Oseen system This matrix-valued function -we denote it by O (see Section 2 for its definition)is completely explicit, in the sense that each entry of O(x), for x ∈ R 3 \{0}, consists of sums, products and quotients of x 1 , x 2 , x 3 , |x|, e −τ (|x|−x1)/2 and τ . No smallness conditions are involved in the purely translational case. For the flow considered here -around a body performing a rotation and a translation -, only partial results are known. They are due to Kyed [61], who considered a leading term given by the product of a coefficient times a column of O. He showed that the corresponding remainder R(x) belongs to L p -spaces in the complement B c S of a ball B S := {y ∈ R 3 : |y| < S} containing D, where the parameter p > 1 can be chosen so close to 1 that the function |x| s τ (x) −1 is excluded from L p (B c S ). This indicates but does not prove that the remainder might decay faster than |x| s τ (x) −1 for |x| → ∞, that is, faster than the rate given by (3) for u(x). Using a similar reasoning, the author suggests that the gradient of his remainder may exhibit a more rapid pointwise decay than the one given by (3) for ∇u(x). Again the question remains open whether this is actually the case. In [62], Kyed indicates that |R(x)| behaves as O(|x| −4/3+ ) if |x| → ∞, for some arbitrary but fixed > 0.
As a further restrictions, the theory in [61] only covers Leray solutions satisfying a no-slip boundary condition on ∂D, and it requires f = 0. Concerning the argument adopted in [61] in order to derive these results, it is based on certain estimates of solutions to the time-periodic Oseen system in the whole space R 3 . These estimates, in turn, are obtained via Fourier expansions of these solutions, an approach which ultimately reduces to a study of solutions to the standard Oseen system (5) in the whole space R 3 .
It is the purpose of the work at hand and of a companion paper [12] to present an improved theory on a leading term associated with problem (1), (2). Our aim is to bring this theory up to the level attained in the purely translational case ( = 0). More precisely, we will study Leray solutions to (1), (2) without requiring any boundary condition on ∂D.
The leading term we will arrive at is of the same kind as the one in [61], that is, a coefficient times a column of O, which means in particular that the rotation does not contribute anything to that term. We will show that the coefficient in question depends in an explicit way on the restriction of u, ∇u and π to ∂D. (At the end of this section, we comment on the regularity issues this situation raises.) If the boundary condition imposed in [61] is satisfied, our coefficient coincides with the one in that reference. This was announced in [14] and proved in [12].
Moreover -and this is the focus of the work at hand -we will establish optimal pointwise decay rates for the remainder term and its gradient (Theorem 3.1, which is the main result of this article). Thus we are able to answer the questions left open in [61], filling a major gap in the theory of flows around rotating bodies.
The approach we will use in the following and in [12] does not start with the splitting of u we actually have in mind. Instead we will first construct a leading term based on the matrix-valued velocity part, denoted by Z(x, y) for x, y ∈ R 3 , x = y, of a fundamental solution to the linearized system This function, proposed in [67] (see Section 2 for its definition), has the inconvenient feature that each of its entries is defined via an integral on (0, ∞), and thus is not explicit at all. So this function is not suitable to come up in a leading term. In spite of that, we will begin by considering the term L(x) : 3 being defined via restrictions of u, ∇u and π to ∂D, where the pair (u, π) is a given Leray solution of (1), (2). (As we may recall, we will comment further below on the regularity issues this definition raises.) The principal result of the present article (Theorem 3.1) then states that the remainder R j (x) := u j (x) − L j (x) and its gradient decay with an optimal rate for |x| → ∞.
This leaves open the question how to get from the leading term 3 k=1 b k Z jk (x, 0) to another one involving O(x). The answer is given in the companion paper [12], where it is shown that the coefficients in our leading terms and coefficients in [61], coincide if we take into account also the boundary conditions. In [12] we deal with the integral on (0, ∞) appearing in each entry of Z(x, 0). For the entries in the first column of Z(x, 0), we will be able to compute the integral in question, obtaining the first column of O(x), which, after multiplication by b 1 , corresponds to the leading term we actually look for. As concerns the other two columns, the integral appearing in their entries turns out to be oscillating, and therefore may be shown to decay more rapidly -as a function of x -than the right-hand side of (3). As a consequence, we may subsume these entries in the remainder term, thus completing our argument.
Let us briefly indicate what is the principal technical difficulty we have to deal with in the work at hand. Recall that we intend to estimate the remainder R(x) and its gradient for large values of |x|. To this end we will start with an integral representation of u(x) proved in [7] (weak solutions) and in [11] (Leray solutions), and restated below as Theorem 2.15. One of the integrals appearing in this representation is the volume potential D c Z(x, y) [(u · ∇)u](y) dy. When we look for a bound of the gradient of the remainder, this integral gives rise to the problem of estimating the term D c ∂ x l ∂ xm Z(x, y) [(u · ∇)u](y) dy , for x from outside a ball B S , and for 1 ≤ l, m ≤ 3. The factor [(u · ∇)u](y) in this term can be handled by using (3). As concerns the second derivatives ∂ x l ∂ xm Z(x, y) of Z(x, y), however, no suitable bounds are available in literature, so we have to return to the definition of Z(x, y), which, as we may recall, involves an integral on (0, ∞). In this way we arrive at an integral on D c × (0, ∞), which we split into several parts. Each of these parts will then be estimated separately (proof of Theorem 3.1). We mentioned above that the coefficients b j (1 ≤ j ≤ 3) in our leading term are defined via restrictions of u, ∇u and π to ∂D. Of course, this is feasible only if the boundary ∂D exhibits some regularity, and if ∇u and π are somewhat more regular near ∂D than required for Leray solutions. This situation is of interest when the link between boundary conditions and the asymptotic of u is to be studied. However, our theory does not depend on any additional regularity of u or π beyond what is required for Leray solutions. To see this, suppose that f has compact support, as we will do in the following, and choose S > 0 large enough so that supp(f ) and D are contained in the ball B S . Then it is an easy consequence of interior regularity of solutions to the Stokes system that both u and π are of class C ∞ outside B S c (proof of Corollary 2). Hence we may consider the pair of functions (u|B 2S c , π|B 2S c ), say, instead of (u, π), thus avoiding any assumptions on ∂D and any additional conditions on u and π (Corollary 2). Concerning further articles related to the work at hand, we mention the fundamental paper [6], where the representation formula (Theorem 2.15) is derived. For the properties of the representation formula see [8]- [10], [13]. With respect to the purely rotational linearized case in an L q framework, we refer to the work of Farwig et al. see [23]. For an approach with the Kondratiev type weight functions, see [1]. A case where also translation is included see [16]. Concerning nonlinear L q setting we refer to the work of Farwig, Hishida [19], Heck et al. [50]. A case with nondecaying initial data was studied by Giga et al. [43]. The spectrum of the "rotating" Stokes operator was studied by Farwig and Neustupa [26]. A weighted analogue can be found in the work of Farwig et al. see [24,25]. Concerning the physical motivation, more references and fundamental theory in the L 2 framework, see [34], [35]. Let us mention the work of Hishida who proved that semigroup generated by the Stokes operator with rotating terms is not analytical. The extention to L p case can be found in the work of Hieber et al. [42]. L p − L q estimates were studied by Hishida and Shibata [48] and [49]. An investigation of Leray solution can be found [38]- [40]. Anisotropic weights in an L 2 or L q framework were studied in [52]- [55]. An asymptotic profile of both linearized and nonlinear problems were exhibited in [59]- [61]. A different subject is rotating fluids where the Coriolis force or centrifugal force play important role. In this point let us mention the work of Feireisl et al. [30], [31], where singular limits are studied. Also the case of rotating fluids was studied for rough boundaries [44].

Notation and preliminaries.
The open bounded set D ⊂ R 3 introduced in Section 1 will be kept fixed throughout. We assume its boundary ∂D to be of class C 2 , and we denote its outward unit normal by n (D) . The numbers τ and and the vector ω also introduced in Section 1 will be kept fixed, too. Define the matrix Ω ∈ R 3×3 by We recall that the function s τ was defined in Section 1, as was the notation |α| for the length of a multi-index α ∈ N 3 0 .
We begin by introducing the fundamental solutions used in what follows. We set where Γ denotes the usual Gamma function. In the following, the letter Γ will stand for the matrix-valued function defined by (Γ jk (y, z, t)) 1≤j,k≤3 := (Λ rs (y − τ t e 1 − e −tΩ · z, t)) 1≤r,s≤3 · e −tΩ , We will use the ensuing estimate, which was proved in [15].
Moreover, for α, β as before, the derivative ∂ α y ∂ β z Z jk (y, z) is a continuous function of y, z ∈ R 3 with y = z.
Due to Lemma 2.5, this means for y, z as above, and for j, k ∈ {1, 2, 3}, α ∈ N 3 0 with |α| ≤ 1 that Then F ∈ C 1 (D c ) and ∂ z l Z jk (y, z) g(z) dz for y ∈ B R c .

PAUL DEURING, STANISLAV KRAČMAR ANDŠÁRKA NEČASOVÁ
Then F ∈ C 1 (B R c ) and Finally, we need some pointwise estimates of convolutions of the type A logarithmic factor on the right-hand side of these estimates is also admitted. For a systematic study of estimates of the convolutions of the type η −a −b * η −c −d of for all real values of parameters a, b, c, d, we refer to [15,56]. This type of estimate will be needed here in two concrete cases: . Then there is a constant C(γ) > 0 such that for all x ∈ R 3 : Lemma 2.13. There exist a constant C > 0 such that for all x ∈ R 3 : Starting point of our considerations will be the following theorem about the integrability and pointwise decays of the velocity and its gradient, where by the velocity we mean the velocity part of a solution is to the rotational Navier-Stokes equations: Theorem 2.14 ([10, Theorem 1.1]). Let τ ∈ (0, ∞), ω ∈ R 3 \{0}, D ⊂ R 3 open and bounded. Take γ, S 1 ∈ (0, ∞), with the constant D depending on τ, ρ, γ, S 1 , p 0 , A, B, f |B S1 1 , u, π, S, and on an arbitrary but fixed number S 0 ∈ (0, S 1 ) with D ⊂ B S0 .
For y ∈ R 3 , j ∈ {1, 2, 3}, we set According to [7, Lemma 3.1], the integral appearing in the definition of R j (f ) is well defined at least for almost every y ∈ R 3 . If f is a function on D c , the function f in the previous definition is to be replaced by the extension of f by zero to R 3 .
In order to derive the leading terms of the velocity and its gradient we are going to use a representation formula of a solution of the rotational Navier-Stokes equation: for T as above.
Suppose that the pair (u, π) is a weak solution of the Navier-Stokes system with Oseen and rotational terms, and with right-hand side f in the sense of (12). Then where B j (u, π) is defined by Proof. See [11,Theorem 4.1], and its proof, as well as [7,Theorem 4.4].
In comparison with the linear case we will need some additional lemma: Proof. Indeed: i.e. ∇ z (φ(Az)) = A T ∇φ(Az), which gives the mentioned formula.
Corollary 1. In the situation of Theorem 2.14, we get for z ∈ B S1 c that   (16).
Proof. Let U ⊂ R 3 be open and bounded, with U ⊂ B S1 c . It is enough to show that (16) holds for x ∈ U , that V|U ∈ C 1 (U ) 3 , and (17) is valid for x ∈ U . Due to our assumptions on U , we may choose R, S ∈ (S 1 , ∞) such that B S ∩U = ∅ and U ⊂ B R . In particular we have dist(B S , U ) > 0 and dist(U, B c R ) > 0. This observation and Lemma 2.8 imply that |∂ α x Z(x, y)| ≤ C 0 for x ∈ U, y ∈ B S \D, α ∈ N 3 0 with |α| ≤ 1, where C 0 is independent of x and y. We further observe that (u · ∇)u ∈ L 3/2 (D c ) 3 , hence (u · ∇)u|B S \D ∈ L 1 (B S \D) 3 . Lemma 2.8 and (13) yield that |∂ α x Z(x, y) [(u · ∇)u](y)| ≤ C 1 · |y| −7/2−|α|/2 for x ∈ U, y ∈ B c R , with C 1 again being independent of x and y. In view of the last statement of Lemma 2.7, we may thus conclude by Lebesgue's theorem that the function is integrable for x ∈ U, α ∈ N 3 0 with |α| ≤ 1, that the function belongs to C 1 (U ) 3 , and that ∂ α V (I) (x) = A ∂ α x Z(x, y) [(u · ∇)u](y) dy for x, α as before.
Using Lemma 2.9 and Theorem 2.14, we see there are constants C 2 , C 3 with x ∈ U, is continuously differentiable, as follows from Lemma 2.6. Now we may conclude from Lebesgue's theorem that the function y → ∂ α x Z(x, y) ϕ δ (x − y) [(u · ∇)u](y), y ∈ B R \B S , is integrable for any δ > 0, x ∈ U, α ∈ N 3 0 with |α| ≤ 1, the function belongs to C 1 (U ) 3 for any δ > 0, and for δ, x, α as before. Proceeding as in (18), we further obtain for x ∈ U, α ∈ N 3 0 with |α| ≤ 1, with C 4 , C 5 denoting constants independent of δ and x. Therefore, by an argument involving uniform convergence of B δ and ∇B δ for δ ↓ 0, we may conclude that the function the proof of the lemma is complete.
3. Leading term of the velocity and of its gradient. The aim of this part is to find the leading term of the velocity and its gradient for the Navier-Stokes problem with rotational terms. Let us recall that the quantities τ, ω and the set D were fixed in Section 2. We study the case f has a compact support in D c . The result we will prove in the work at hand may be stated as: 9 and u|∂D ∈ W 2−1/p, p (∂D) 3 , π ∈ L 2 loc (D c ) with π|D S1 ∈ L p (D S1 ). Suppose that the pair (u, π) is a weak solution of the Navier-Stokes system with Oseen and rotational terms, and with right-hand side f in the sense of (12). Then there are coefficients β 1 , β 2 , β 3 ∈ R and functions F 1 , F 2 , F 3 ∈ C 1 (B S1 c ) such that for j ∈ {1, 2, 3}, α ∈ N 3 0 with |α| ≤ 1, x ∈ B S1 c , where where C depends on τ, ω, p, S 1 , S, certain norms of u, π and f , and on the constant D from (13).
In Theorem 3.1, the estimate presented in [11,Theorem 3.14] for the linear case is extended to the nonlinear one. Note that by [67, (3.9)], the function Z(x, 0) in the leading term on the right-hand side of (19) corresponds to the time integral of a fundamental solution of the evolutionary Oseen system multiplied by a rotation depending on time.

Remark 2.
In the case of [61], [62] the term is zero because of the boundary conditions.
In other words, the decay rates of the remainder and its gradient are optimal in the sense that they correspond to those of an additional derivative of O, apart from a logarithmic factor. A comparison with O seems to be reasonable in view of the fact that we want to arrive at a leading term of the form coefficient times a column of O.
Proof of Theorem 3.1. The term of (20) contained in braces {. . . } we will be called "the leading term", term F we will call "the remainder". From Theorem 2.15 we have where B j (u, π) was defined in (15). We put By the definition of β k the leading term in formula (19) is determined. Because (19) is in fact rearrangement of formula (22), we now define the value F j as the difference of the right-hand side of the representation formula (22) minus the leading term. We will distinguish F (I) coming from the linear terms and F (II) arising from the non-linear part, i.e. from R j (u · ∇)u : for x ∈ B S1 c , 1 ≤ j ≤ 3. Then by (22) we get (19).
2. C 1 -continuity of F II . By Lemma 2.17, the function F (II) ∈ C 1 (B S1 c ) 3 , and first-order derivatives may be moved into the volume integral appearing on the right-hand side of (23).

Estimates of
we apply firstly the integration by parts and and then split the resulting volume integral in an integral B R on the bounded domain B R \ D and integral E R on the Of course, here and in similar situations in the following, a partial integration has to be performed first on a bounded domain, where B T \(B R ∪ B (x)) with T > max{2R, 2|x|}, 0 < is a good choice for such a domain. In the next step we let tend to zero. This passage to the limit may be handled by referring to Lemma 2.9 and (13). Finally we let T tend to infinity. The surface integral on ∂B T which came up in the partial integration then vanishes, as follows from Lemma 2.8 and (13). The same references imply that all the volume integrals involved tend to integrals on B c R when T → ∞. In volume integral E R over the exterior domain (B R ) c we use firstly the definition of Z, (9) and the Fubini's theorem, and then the domain invariant transformation y = e tΩ z for fixed t > 0. The reason why we use the mentioned transformation is that we would like to avoid a periodic term in the right-hand side of (30): Finally we split the the exterior domain of integration B c R on two domains: Let δ be a sufficiently small positive number comparing to 1, R and S − S 1 , f.e. δ := min{1, ( We obtain: Substituting the expression of E R into (25) we get: ∂ α x B R , ∂ α x S ∂D : Estimating the first two terms and their derivatives ∂ α x for |α| = 0, 1, we get the following estimate: Indeed, from Lemma 2.8 for y ∈ B R , x ∈ B c S : for some 0 ≤ θ ≤ 1. So, with Lemma 2.11 and (28) Similarly, we have with (10) and (29): For the estimate of this term we use Lemma 2.5 for the first order derivatives of Γ : We have (for x = e tΩ z) From Theorem 2.14 we have for y ∈ B c R |u(y)| 2 ≤ C(R) (|y| s τ (y)) −2 .
If z ∈ (B R ) c then e tΩ z ∈ (B R ) c , we get: Since B δ (x) ⊂ B c R , we thus get due to (6), So we have where the integral with respect to variable t is estimated using Lemma 2.9, choosing in its application y := x − z, z := 0. V R,δ : Similarly as in the previous case, using (30) and (31)we find Now, the integral with respect to t can be estimated using Lemma 2.6 with y := x − z, z := 0.
The last inequality follows from lemma 2.12 with γ = 2.

5.
Estimates of ∂ α F (II) for |α| = 1. Let us mention that S, S 1 , R, δ are the same as in the previous section, so The aim of this part is to find the leading term of the gradient of velocity for the Navier-Stokes problem with rotational; terms: Unlike in the previous part is that we cannot apply an integration by parts over the whole domain D c because we have to exclude the neighbourhood B δ (x) due to singularities of the second order derivatives of Z. On the other hand, to avoid some technical difficulties, we are able to handle the integrals with respect to t only in domains invariant with respect to the transformation y = e tΩ z, t > 0. These facts cause some additional computations. So we use Lemma 2.17, split the domain of integration into the bounded part B R \ D c and the exterior domain (B R ) c , and then perform an integration by parts firstly only on the bounded domain: So we get: Concerning the last term in (34), we find due to (9) that ∂ α x Γ jk (x, y, t) (u l ∂ l u k ) (y)dy dt The domain of integration of E R is (B R ) c . This exterior domain is invariant with respect to the transformation y = e tΩ z, t > 0. We may use the same transformation to avoid periodic terms as in the case |α| = 0 : Unlike in the case |α| = 0, this transformation is used before the integration by parts. More precisely, we split the domain of integration into the two domains B δ (x) and (B R ) c \ B δ (x). In the integral over the unbounded domain we apply the identity from Corollary 1 and integrate by parts: ∂ α x Γ jk (x, e τ Ω z, t) (u l ∂ l u k ) (e τ Ω z)dz dt ∂ α x Γ jk (x, e tΩ z, t) (u l u k ) (e tΩ z) e tΩ (x − z)/δ l do z dt ∂ α x Γ jk (x, e tΩ z, t) (u l u k ) (e tΩ z) e tΩ (−z)/R l do z dt e tΩ ∇ z l ∂ α x Γ jk (x, e tΩ z, t) (u l u k ) (e tΩ z) dz dt = U δ j + S δ j + − S R j + U R,δ j Substituting the expression of E R (x) into (34) and using (9), we get finally: Now we will estimate all terms of (35) for |α| = 1: ∂ α S ∂D , ∂ α B B R : From (27) we know that |∂ α x S ∂D | + |∂ α x B R | ≤ C 1 (S 1 , S) (|x| s τ (x)) −2 .
It is clear that e tΩ (x − z)/δ = 1 for z ∈ B δ (x). So we have where the integral with respect to variable t is estimated using Lemma 2.9 with y := x − z, z := 0. So, the integral S δ may be subsumed into to the remainder.
The last inequality follows from Lemma 2.13.
Remark 3. So, we finally get that the leading term of ∂ α u j takes the same form as in the linear case, that is 3 k=1 β k ∂ α x Z jk (x, 0), but with the definition of β = containing the additional term ∂D 3 j=1 ((n (D) ) j u j u)(y) do y .
Therefore, by interior regularity of the Stokes system, as stated in [