On the asymptotic properties for stationary solutions to the Navier-Stokes equations

In this paper we study solutions of the stationary Navier-Stokes system, and investigate the minimal decay rate for a nontrivial velocity field at infinity in outside of an obstacle. We prove that in an exterior domain if a solution \begin{document}$ v $\end{document} and its derivatives decay like \begin{document}$ O(|x|^{-k}) $\end{document} for sufficiently large \begin{document}$ k $\end{document} , depending on the velocity field, as \begin{document}$ |x|\to \infty $\end{document} , then \begin{document}$ v $\end{document} is zero on that exterior domain. Constructive estimate for \begin{document}$ k $\end{document} is given. In the case where velocity field is only bounded at infinity, we show that the infimum of \begin{document}$ L^2 $\end{document} norm of a velocity field on a unit ball located at distance \begin{document}$ t $\end{document} from an origin is bounded from below as \begin{document}$ Ce^{-\beta t^\frac 43\ln(t)}. $\end{document} The proof of these results are based on the Carleman type estimates, and also the Kelvin transform.


Introduction.
Let Ω be a simply connected open, bounded subset of R 3 containing the origin. We consider the stationary Navier-Stokes equations in an exterior domain R 3 \ Ω: where v = (v 1 , v 2 , v 3 ) = v(x) and p = p(x), are the velocity and the scalar pressure of the fluid flows respectively. In the following we consider problem of behaviors at spatial infinity of a nontrivial solutions of (1) in the exterior domain, R 3 \ Ω. The first result in this direction was obtained by Finn [5], who showed that a velocity field satisfying the zero Dirichlet boundary condition on ∂Ω, which decays at spatial infinity as o(|x| −1 ), is trivial. In the case without any boundary condition on ∂Ω, Dyer and Edmunds [4] proved that any C 2 bounded velocity field decaying at spatial infinity as O(exp(− exp(s|x| 3 ))) for any positive s is trivial. One can construct nontrivial solution of the (1) in any exterior domain, having a zero of any sufficiently large order at spatial infinity. Indeed, given k ≥ 1 let us set φ k (x) = ∂ k x1 1 |x| .
In the theorem below we try to estimate the decay rate on infinity for nontrivial, non-gradient solutions to (1) based on some properties of this solution. Below we denote B(0, r) = {x ∈ R 3 | |x| < r}, S(0, r) = {x ∈ R 3 | |x| = r}.
We have Then γ ∈ (0, +∞) and solution v can not have zero at spatial infinity of order s * greater then γ + 1.
Proof of theorem 1.2 is based on the analysis of the vorticity equation (66), using Carleman estimate with boundary with the logarithmic weight function. Decay assumption (4) on velocity field allow us to treat first order terms in the vorticity equation (66) as a perturbation and use factorization of the Laplace operator into product of two first order operators.
This function was introduced in paper [1] to study the localization effect for the Schrödinger equation in R n . In particular they proved that the minimal decay rate on infinity for nontrivial solution is M (t) ≥ Ce −βt 4 3 ln t with some positive β. Several efforts have been done to extend such a result for the stationary Navier-Stokes and the Stokes systems. In [11] the authors proved that the decay rate for M (t) is greater then double exponent, and in [12] they improved the estimate M (t) ≥Ce −Ct 2 ln t for some positive C,C. In this paper this result is improved in the following two theorems: In order to formulate the first theorem we introduce parameterγ : γ =γ(v) = lim t→+∞ Λv L 2 (S(0,t)) / v L 2 (S(0,t)) .
We have Then there exist k > 0 depending on v and a constant t 0 depending on v such that If curl v not identically equal to zero, then in inequality (7) one can take where µ is the maximal root of the polynomial 16λ 2 + 8λ + 11 − 4γ 2 and parameter γ defined by (3). If curl v ≡ 0 in inequality (7) one can take k defined by (8) where µ is the maximal root of the polynomial 16λ 2 + 8λ + 11 − 4γ 2 and parameterγ defined by (5).
For our next result we make no assumptions on decay of the velocity field on infinity.
The estimate (10) proved in theorem 1.4 improves the corresponding estimate from [11] . The machinery used in the proof is the reduction of the stationary Navier-Stokes system to the Shrödinger equation (136). This allows us apply approach developed in [1]. Results of [1] can be applied directly if we consider the Navier-Stokes system in the whole space R 3 . In can when Ω is nonempty bounded set the technique of [1] requires some modification (see Appendix). Assumption on boundedness of domain Ω is essential. If domain Ω is unbounded the counterexample to estimate (10) can be easily constructed. The estimate obtained in theorem 1.3 is stronger then estimate (10). The main reason for the improvement is the following. Proof is based on the Carleman estimates with boundary and the decay assumption on the vector field (6) allow us to use ln |x| as a weight function in the Carleman estimate. The choice of such a function under assumptions of theorem 1.4 is impossible.
The following theorem states the asymptotic behavior of the pressure of the Navier-Stokes equations.
Then there exist positive constants C 1 = C 1 (v) and C 2 = C 2 (v) such that 2. The Carleman estimates. In this section we obtain a Carleman type estimate for use in the proof of the main theorems of the previous section. A Carleman type estimates originally were introduced in [3] by T. Carleman in order to prove uniqueness of solutions for Cauchy problems for elliptic equations. Later Carleman estimates we obtained for general scalar partial differential equations under condition of existence of a pseudoconvex functions (see e.g. [8], [9]) and under more restrictive conditions Carleman estimates were obtained for systems of partial differential equations (see e.g. [2], [8]). Recently it was realized that Carleman estimates are very effective tool for proving observability estimates for solutions of evolution equations and consequently for obtaining exact boundary controllability results for evolution equations. This theory requires so-called Carleman estimates with boundary (survey of recent results can be found in [6].) In this section we will prove a Carleman estimate with boundary and with singular weight function. The main Carleman estimate is obtained in lemma 2.2 for the scalar Schrödinger equation. In proposition 1 this estimate is generalized for the elliptic system with the first order terms. Lemma 2.3 and proposition 2 are used in the proof of theorem 1.2 and proposition 5 is used in the proof of theorem 1.3. Proposition 3 and proposition 4 are of independent interest.
We also set L + (x, s, D)u := 1 2 (L(x, s, D) +L(x, s, D))u = ∆u + 4s + 4s 2 |x| 2 u, Definition 2.1. We say that a function w ∈ H 2 (Ω) has a zero of order s 0 at the point x = 0 if there exists the following limit, lim r→+0 Given a bounded domain Ω, we define ρ = ρ(Ω) as follows Our main purpose in this section is to prove the following: Let Ω be a bounded domain in R 3 with ∂Ω belonging to C 1 class. Suppose w ∈ H 2 (Ω) is a solution of (12), having zero at origin of the order s 0 for some strictly positive s 0 such that s 0 > 5 2 . We set u := |x| −2s w, and write the problem (12) in terms of u as Then, we have the following estimate: for all s ∈ [1, s0 2 − 1 4 ) and ∈ (0, 1) provided that |x| −2s f ∈ L 2 (Ω).
Proof. For all sufficiently small positive δ we set Ω δ = Ω \ B(0, δ). We take the L 2 (Ω δ ) norm of the left and right hand side of (13): Let us compute the last term in this equality. We will repeatedly use the following facts during computations.
We first have Let us calculate the terms I j (δ), j = 1, 2, 3, separately. Therefore, Using (20), we write the equality (15) as Simple computations imply Using (22), we transform (21) into We have We also estimate After division of this inequality by ρ 2 we have From the estimates (23)-(25) we obtain Passing to the limit in (26) as δ → +0 we obtain This estimate implies (14) in the case s ∈ [1, s0 2 − 1 4 ) when the last term in the right hand side is absent.
Next we prove estimate for the last term in the right hand side of (14). For any ∈ (0, 1) we have Setting v = u/|x| 2− we write the integral in the right hand side of the above inequality as Denote g = 2(x, ∇)v + (3 − 2 )v and take the scalar product of this function in L 2 (Ω) with v . After short integration by parts we have Using Young's inequality, −gv ≤ v 2 + 1 4 g 2 , we obtain From (28)-(31) we have This concludes the proof of the lemma for the term s 2 in the right hand side. Taking the scalar product of the function −u/(ρ 2 |x| 2−2 ) with the function L + (x, s, D)u in L 2 (Ω δ ) we have for s ∈ [1, s 0 − 1 4 ) and ε ∈ (0, 1), where we used (32). This implies the inclusion of the last term in right hand side of estimate (14). Remark 1. The Carleman estimate with the weight ln |x| was proved in e.g. in [7] and [14] for the zero Cauchy data on the boundary and in [10] with the Dirichlet boundary conditions for the boundaries which does not contain the origin.
We prove the following: Let Ω be a bounded domain in R 3 with ∂Ω belonging to C 1 class and is a solution of (12), having zero at infinity of the order s 0 > 5/2 and we assume f |x| 2s0 ∈ L 2 (Ω). We set u := |x| 2s w, and write the problem (12) in terms of u as Then, the following hold true. and Passing to the limit as R → +∞, we obtain the equality (40). Setting v = u/|x| 2 , we obtain We take the scalar product of g in L 2 (R 3 \ Ω) with v/ ln |x|. After short integration by parts we have Applying the Young's inequality to (45), we obtain and therefore, using (44) one has By (46) proof of inequality (41) is complete. On the other hand From this inequality we obtain The proof of the proposition is complete.
We also need the Carleman estimate similar to one obtained in proposition 1 for the case when the function u does not satisfy the Dirichlet boundary conditions.

Proposition 2.
Let Ω be a bounded domain in R 3 with ∂Ω belonging to C 1 class and 0 ∈ Ω. Suppose w ∈ H 2 (R 3 \ Ω) is a solution of (12), having zero at infinity of the order s 0 > 5/2. We set u := |x| 2s w. Then, the following holds true.
for all s ∈ [1, s0 2 − 1 4 ) provided that |x| 2s ∆w ∈ L 2 (R 3 \ Ω). Remark 2. In lemma 2.3 and proposition 2 the assumption 0 ∈ Ω is important. Without such an assumption 0 ∈ R 3 \ Ω and we have to assume, in order to prove the equalities (40) and (48), that function u has the zero of an appropriate order at 0. In lemma 2.3 the more stronger assumption, namely B(0, 1) does not belong to Ω, allows us to keep the parameter r 0 positive.
Using the Carleman estimate (14), we first prove the following.

Proposition 3.
Let Ω be a bounded, star-shaped domain respect to the origin in R 3 with ∂Ω belonging to C 1 class, c ∈ L ∞ (Ω). Suppose w is a solution to the boundary value problem If the function w at origin has zero of order Proof. We set u := |x| −2s w. Since the domain Ω assumed to be star-shaped respect to the origin, we have (x, ν) ≥ 0 on ∂Ω.
This implies Let s 0 be such that Then, we apply the estimate (14) to equation (49), This estimate and (50) yields Thanks to (51) this estimates implies that u = w ≡ 0. Proof of the proposition is complete.
Below we apply the above argument to the decay problem in the exterior domain.

Proposition 4.
Let Ω be a bounded domain in R 3 , star-shaped respect to the origin, and the function c = c(x) satisfies c(x)|x| 2 ln |x| ∈ L ∞ (R 3 \ Ω). Let w be a solution to the elliptic equation Suppose w has zero of order s 0 , at spatial infinity, satisfying Then, w = 0 on R 3 \ Ω. Proof.
From this inequality and the choice of s 0 the statement of the proposition follows immediately.
We set The following proposition will be used in the proof of theorem 1.3.
Proof. Repeating the arguments leading to (23) we have For any ∈ (0, 1) we have Setting v = u/|x| 2− we write the integral in the right hand side of (56) as v and take the scalar product of this function in Hence the above inequality combined with (56) and (57) imply (58) Taking the scalar product of the function −u/(R 2 |x| 2−2 ) with L + (x, s, D)u in L 2 (B(0, R)) and using (58) we have From (59) and (58) we obtain (54). Proof of the proposition is complete.
3. Proof of the main results.
Proof of Theorem 1.2. Denote ω = curl v. First we prove that the limit in the right hand side of (3) exists and γ ∈ (0, +∞).
Proposition 6 implies that κ < ∞. We claim that Our proof is by contradiction, suppose that κ = 0. Then By (62) for any positive N > 1 there exists a constant t 0 (N ) such that So by (63) we obtain ω L 2 (S(0,t+2n)) ≤ 1 N n ω L 2 (S(0,t)) ∀t ≥ t 0 (N ). Then for any n ∈ N + and for any t ≥ t 0 (N ) we have We fix positive s and set N = 2 2s and take in (64) t = t 0 (2 2s ). Then inequality (64) has the form Function ω = curl v satisfies the vorticity equation The inequality (65), the assumption (4) and the classical a priori estimates for the Laplace operator imply Letμ(t) be a smooth function such that µ(t) = 1 fort < 0 andμ(t) = 0 fort ≥ 1 2 .
Let t > 1. By (4) and (70) there exists a constant C 3 independent of s and t such that Since the function ω n has a compact support, the Carleman estimate (48) from the proposition 2 applied to vorticity equation (68) holds true for all sufficiently large s in domain R 3 \ B(0, t) with sufficiently large t: Setting v n = u n /|x| 2 , we obtain − 1 2s L − (x, s, D)u n = 2(x, ∇v n ) + 3v n = g n in R 3 \ Ω.
We take the scalar product of g n in L 2 (R 3 \ B(0, t)) with v n / ln |x|. After short integration by parts we have Applying the Young's inequality to (73), we obtain and therefore, using (44) one has On the other hand By (75) and (74) we obtain from (72) that there exists a constant C 4 independent of s such that Using (71) to estimate the first term in the right hand side of (76) we obtain Letx be a point in R 3 such that |x| = 4t + 1. Since B(x,1) |ω| 2 dx = 0 inequality (77) implies (4t) 2s for all sufficiently large s. Passing in inequality (78) to the limit as s → +∞ we obtain that ω| B(x,1) ≡ 0. By the uniqueness of the Cauchy problem for the second order elliptic partial differential equation ω| R 3 \Ω = 0. Proof of (61) is complete.
Proof of (60) is complete.
Using the representation of the Laplace operator in the spherical coordinates as we write equation (66) as where Combining equation (83) with assumption of decay of our solution on infinity implies the existence of a constant C 10 independent of t such that for all sufficiently large t. Indeed taking the scalar product of equation (83) with So passing to the limit in (85) as N → +∞ we obtain This proves estimate (84). Setting w(x) = |x| s * − 1 2 ω(x) we have We divide both sides of equation (86) by |x| and take the L 2 norm of the both sides of equation (86), we obtain the equality Using (84), we estimate the right hand side of (87) to obtain By (4) there exist constants C 12 andt 0 independent of t such that for all t ≥t 0 f |x| s * + 1 2 2 L 2 (R 3 \B(0,t)) ≤ C 12 t |∇w|/|x| + |w|/|x| 2 L 2 (R 3 \B(0,t)) .
Proof of Theorem 1.3. First we consider the case when vorticity ω = curl v is not identically equal zero. Let us introduce the function

OLEG IMANUVILOV
We observe that there existst(v) and a positive constant C 0 independent of t such that In order to prove (91) we introduce the functionρ ∈ C ∞ 0 (B(0, 19 20 )),ρ| B(0, 9 10 ) = 1 and set ρ(x, y) =ρ(x − y) for j ∈ {0, 1}. Obviously there exists a constant C 1 independent of x such that Let W be solution to the following boundary value problem By (92) and classical estimate for the Laplace operator Taking the scalar product of the equation (66)  (2((∇W, ∇ρ 10 (x, ·)), ω) Using the Cauchy inequality, we estimate the right hand side of (95) as From (96) and (94) we obtain the estimate Taking the scalar product of the equation (66) and the function ρ 20 (x, ·)ω we obtain that there exists a constant C 6 independent of x such that Combining (98), (97) we obtain from vorticity equation (66): From (100) and Sobolev embedding theorem we obtain (91).
If the vorticity ω is equal to zero then v is the gradient solution, namely locally there exist a function φ such that v = ∇φ, p = − 1 2 |∇φ| 2 . Since div v ≡ 0 function φ is the harmonic function on a domain of definition. Then all components of the velocity vector v are harmonic functions in R 3 \ Ω. The Kelvin transform u of the function the function v defined by (101) is harmonic in some neighborhood of zero possibly excluding the origin. On the other hand, thanks to our assumption (6) on the decay rate of the velocity field on infinity, the Kelvin transform of the function v is bounded in some neighborhood of origin and therefore is harmonic at this neighborhood. The function u can have at origin the zero of finite order. The inequality (7) with (8) can be obtained repeating the above arguments where instead of equation (102) one use equation ∆u = 0. Proof of theorem is complete.
Proof of Theorem 1.4. First we show that under the assumption of our theorem there exists a constant C 1 such that where B(0, R) is some ball centered at zero of sufficiently large radius such that Ω ⊂⊂ B(0, R − 2). Let W be solution to the following boundary value problem where x ∈ R 3 be a point such that B(x, 2) ⊂ R 3 \ B(0, R) and ω = curl v. By classical estimate for the Laplace operator there exists a constant C 2 independent of x such that Here we used (9) to obtain the last inequality. Let ρ be a smooth function such that ρ ∈ C ∞ (B(0, 2)), ρ| S(0,2) = 0, ρ(y) > 0 ∀y ∈ B(0, 2); ρ(x, ·) = ρ(x − ·).
From (124) for any multi-index α we have This equality and elliptic estimate for the Laplace operator imply that for all k = 1, 2, · · · , where the constant C 21 depends only on r a , r b . Inequalities (126) with k = 2 and (125) imply The inequality (127) combined with (125) implies On the other hand, writing (66) in the form, Observe that this function satisfies to the Schrödinger equation where potential V is given by formula with matrices A, B, C, D, E, F defined by . We note that in order to obtain the last three equations in the system (136) we used the fact that We claim that under the hypothesis (9), we have for sufficiently large R The boundedness in R 3 \ B(0, R) of functions v and ω follows from estimate (133). The fact ∇Φ ∈ L ∞ (R 3 \ B(0, R)) follows from the estimate Hence we established (137) for the function W . Form (133) the boundedness in R 3 \ B(0, R) follows for the components A ij , B ij , C ij , D ij , E ij , F ij of the function V.
In order to show the boundedness for D ij using the equation (134) we estimate +2 ∇v L ∞ (R 3 \B(0,R)) . The first norm in the right hand side of this inequality in finite by (135). The second and third norms are finite thanks to (133) and (115). The proof of (137) is complete.
Let function M 0 (t) be introduced by formula We claim that there exists a positive constant C 29 = C 29 (V, W ) and positive con- The proof of this fact is given in appendix. (We note that for the case Ω = ∅ this inequality first established in [1].) On the other hand there exists a constant C 30 such that Here M 1 (t) is given by (90) and in order to get the last equality we used (91). From (139) and (138) we obtain (10).
Proof of Theorem 1.5. Let us set Φ = p − p 0 + 1 2 |v| 2 :=p + 1 2 |v| 2 . Then, the steady Navier-Stokes system implies that For each a ∈ R 3 and t > 0 we define, By the maximum principle, we have Φ t,a ≤ 0. Multiplying equation (140) by e −s|x| with some positive s and integrating over R 3 , we obtain Let us choose parameter s so that Then, we have Taking into account the estimate which is obvious from (141) for all s greater then one, we obtain from (142) and (143) that Since Φ t,a ≤ 0 on R 3 again, we have We take τ > 6. Let us consider the sequence {x τ } such that We change variables x = y − x τ /2. Then, we observe Using (144), (145) and (146), observing B( xτ 2 , 1) ⊂ R 3 \ B(0, 2) for τ > 6, we have where constant C is independent of s and τ. From this inequality we have We consider two sub-cases. If If Φ 2,0 = 0 on B(0, 1), then this implies Φ = 0 on B(0, 2). Since Φ(x) ≤ 0 on R 3 , and Φ(x) → 0 as |x| → ∞, by the maximum principle applied to L v (x, D)Φ = |ω| 2 we obtain Φ = 0 on R 3 . Hence, ω = 0 on R 3 . Then v = ∇g, where g is a harmonic function in R 3 . But since |v(x)| → 0 as |x| → +∞ we have that g = constant, and therefore v = 0. Proof of theorem is complete.
Appendix. We give the proof of inequality (138).
Proof. Without a loss of generality we may assume that 0 ∈ Ω and Ω ⊂ B(0, 1 20 ). Let us choose constantβ such that Our proof of (138) is by contradiction. Suppose that (138) is false. Then for anỹ β satisfying (147) there exist a constant C and a sequence t j → +∞ in general depending onβ such that M 0 (t j ) ≤ Ce −βt The short computations imply .

This inequality and standard estimates for elliptic equations imply that there exists a sequence
We set χ 0 (x) be a cut-off function such that For any R > 2 letχ R (x) ∈ C ∞ 0 (B(0, 1)) be cut-off function such that Consider the functions W Rj (x) =W ((R j −1)x+z j ), V Rj (x) =Ṽ ((R j −1)x+z j ). The function W Rj satisfies ∆W Rj + (R j − 1) 2 V Rj W Rj = 0 on B(0, 1).
Since there exists a function G ∈ C 0 (B(0, 1)) such that div (ϕ rr ∇ϕ) = 1 there exists a constant C 2 independent of s such that By (155) and (156) for all s ≥s 0 .
We set in the Carleman estimate (157) s = s j = βR 4 3 j where parameter β such that β ∈ max 64 By (147) such β exists and since β > 1 for all sufficiently large j we have s j > s 0 . Then by (147) and (158) for all R j ≥ 10 and for all x ∈ B(0, 1) Using definition of the function f s we have +2 2s∆ϕu + ([χ Rj , ∆]W Rj )e sϕ 2 L 2 (B(0,1)) . By (157), (159) and the above inequality we obtain for all sufficiently large R j there exist a positive constant C 3 independent of j such that Hence by (149), (150) and (160) for all R j ≥ R 0 > 0 and some positive constant C 4 independent of R j such that Moreover observe that B( zj Rj ( 3 Rj − 1), 1 Rj ) ⊂ B(0, 1 − 1 Rj ). This follows from the inequality Therefore by (149) and (150) we note that for all sufficiently large R j .
Next we claim that Function ϕ depends only on |x|, and it is strictly monotone decreasing function of |x| on interval (0, 1). Therefore function ϕ will reach minimum on ball B( On the other hand pointx = Estimating norm of the function W Rj e sj ϕ in the right hand side of (164) by the right hand side of inequality (161) we obtain: By the Taylor's formula for all sufficiently large R j we have Using this equality in (165) and the fact that for all sufficiently large R j By (151) and (158)  Applying this inequality to estimate the last term in the right hand side of (166) for all sufficiently large R j we obtain that W L 2 B yj , From the sequence y j we take a subsequence {y j k } ∞ k=1 which is convergent to some pointŷ such that |ŷ| = 4. Then by (167) functionW is identically equal to zero on the ball B(ŷ, 1). Hence, uniqueness of solution of the Cauchy problem for the second order elliptic equation implies W ≡ 0. We arrived to the contradiction since velocity field v is not identilally equal zero. Proof of inequality (138) is complete. (|∇(µω)| 2 + |(µω)| 2 )dx (168) for all t ≥ t 1 .
Here P * (t, x, D) is the operator formally adjoint to P (t, x, D) and the operator P (t, x, D) is obtained from the operator P (x, D) determined by formula (66) by change of variables x → tx. Setting ω t (x) = ω(tx), v t (x) = v(tx) from (66) we have Instead of the problem (169) consider the following boundary value problem R(t, x, D)u = q * in B(0, 1), u| S(0,1) = 0.
The operator R(t, x, D) has the form R(t, x, D) = ∆ + It obtained from the operator P (t, x, D) in the following way. First we take the inversion change of variables x → x |x| 2 in the operator P (t, x, D). We denote the corresponding operator as Q(t, x, D). Then R(t, x, D) = 1 |x| 5 Q(t, x, D) • |x|. By our assumptions (4) on decay of the velocity field v there exist constant C 0 ( ) independent of t, x and constant t 1 such that for any j ∈ {1, 2, 3} we have |C(t, x)| ≤ C 0 ( ) |x| 2− t , |B j (t, x)| ≤ C 0 ( ) |x| 1− t ∀x ∈ B(0, 1) and t ≥ t 1 .
The first and zero order terms of the operator R(t, x, D) we denote as R 1 (t, x, D). We claim that for any positive δ there exist 0 > 0 and t 2 such that In order to prove (173) we estimate the contribution of first order terms and zero order terms to the norm the operator R 1 (t, x, D) separately. Let 0 = 0 (δ) ∈ (0, 3 2(2− ) ). Using (172) we have Cw where constant C 9 is independent of t and q * . Next we take the Kelvin transform of problem (171) to obtain a solution to problem (169). Take q = ω t and denote the corresponding solution to problem (170) as w t . By (177) the following estimate is true ∂ r w t L 2 (S(0,1)) ≤ C 10 ω t χ B(0,3)\B(0,1) L where constant C 10 is independent of t provided that t ≥ t 3 .
The vorticity equation (66) and standard estimates for elliptic equation imply the existence of a constant C 18 independent of t such that (|∇ω| 2 + |ω| 2 )dx.