Optimal three-ball inequalities and quantitative uniqueness for the Stokes system

In this paper we study the local behavior of a solution to the Stokes system with singular coefficients. One of the main results is the bound on the vanishing order of a nontrivial solution to the Stokes system, which is a quantitative version of the strong unique continuation property. Our proof relies on some delicate Carleman-type estimates. We first use these estimates to derive crucial \emph{optimal} three-ball inequalities. Taking advantage of the optimality, we then derive an upper bound on the vanishing order of any nontrivial solution to the Stokes system from those three-ball inequalities.


1.
Introduction. Assume that Ω is a connected open set containing 0 in R n with n = 2, 3. In this paper we are interested in the local behavior of u satisfying the following Stokes system: ∆u + A(x) · ∇u + B(x)u + ∇p = 0 in Ω, ∇ · u = 0 in Ω, (1.1) For the Stokes system (1.1) with essentially bounded coefficients A(x), the weak unique continuation property has been shown by Fabre and Lebeau [6]. On the other hand, when A(x) satisfies |A(x)| = O(|x| −1+ǫ ) with ǫ > 0, the strong unique continuation property was proved by Regbaoui [20]. The results in [6] and [20] concern only the qualitative unique continuation theorem and both results require the vanishing property for u and p. In this work we aim to derive a quantitative estimate of the strong unique continuation for u satisfying (1.1) with an appropriate p.
For the second order elliptic operator, using Carleman or frequency functions methods, quantitative estimates of the strong unique continuation (in the form of doubling inequality) under different assumptions on coefficients were derived in [4], [7], [8], [15], [17]. For the power of Laplacian, a quantitative estimate was obtained in [18]. We refer to [17] and references therein for development in this direction.
Since there is no equation for p in the Stokes system (1.1), we apply the curl operator ∇× on the first equation and obtain ∆q + ∇ · F = 0, (1.3) where q = ∇ × u is the vorticity and for n = 2, ∇ × u = ∂ 1 u 2 − ∂ 2 u 1 . For n = 3, ∇ · F is a vector function defined by (1.4) When n = 2, ∇ · F is a scalar and we simply drop the suffix i in the definition above. Now we define ∇ ⊥ × G = ∇ × G for any three-dimensional vector function G and ∇ ⊥ × g = (∂ 2 g, −∂ 1 g) for a scalar function g if n = 2. It is easy to check that ∆u = ∇(∇ · u) − ∇ ⊥ × (∇ × u) and thus we have ∆u + ∇ ⊥ × q = 0 (1.5) if ∇ · u = 0. However, equations (1.3) and (1.5) do not give us a decoupled system. The frequency functions method does not seem to work in this case. So we prove our results using Carleman inequalities. On the other hand, since the coefficient A(x) is more singular than the one considered in [20]. Carleman-type estimates derived in [20] can not be applied to the case here. Hence we need to derive new Carleman-type estimates for our purpose. The key is to use weights which are slightly less singular than the negative powers of |x| (see estimates (2.4) and (2.15)). The estimate (2.15) is to handle (1.3) and the idea is due to Fabre and Lebeau [6]. We can derive certain three-ball inequalities which are optimal in the sense explained in [5] using (2.4) and (2.15). We would like to remark that usually the three-ball inequality can be regarded as the quantitative estimate of the weak unique continuation property. However, when the three-ball inequality is optimal, one is able to deduce the strong unique continuation from it. It seems reasonable to expect that one could derive a bound on the vanishing order of a nontrivial solution from the optimal three-ball inequality. A recent result by Bourgain and Kenig [3] (more precisely, Kenig's lecture notes for 2006 CNA Summer School [14]) indicates that this is indeed possible, at least for the Schrödinger operator. In this paper, we show that by the optimal three-ball inequality, we can obtain a bound on the vanishing order of a nontrivial solution to (1.1) containing "nearly" optimal singular coefficients. Finally, we would like to mention that quantitative estimates of the strong unique continuation are useful in studying the nodal sets of solutions for elliptic or parabolic equations [4], [9], [16], or the inverse problem [1].
We now state the main results of this paper. Their proofs will be given in the subsequent sections. Assume that there exists 0 < R 0 ≤ 1 such that B R0 ⊂ Ω. Hereafter B r denotes an open ball of radius r > 0 centered at the origin. Theorem 1.1. There exists a positive numberR < 1, depending only on n, such where the constant C depends on R 2 /R 3 , n, and 0 < τ < 1 depends on R 1 /R 3 , R 2 /R 3 , n. Moreover, for fixed R 2 and R 3 , the exponent τ behaves like 1/(− log R 1 ) when R 1 is sufficiently small.

Remark 1.2.
It is important to emphasize that C is independent of R 1 and τ has the asymptotic (− log R 1 ) −1 . These facts are crucial in deriving an vanishing order of a nontrivial (u, p) to (1.1). Due to the behavior of τ , the three-ball inequality is called optimal [5].
It should be emphasized that three-ball inequalities (1.6) involve only the velocity field u. This is important in the application to inverse problems for the Stokes system, for example, see [2] and [10]. Using (1.6), we can also derive an upper bound of the vanishing order for any nontrivial u satisfying (1.1), which is a quantitative form of the strong unique continuation property for u. Let us now pick any R 2 < R 3 such that R 3 ≤ R 0 and R 2 /R 3 <R. for all R with R < R 2 .
Remark 1.4. Based on Theorem 1.1, the constants K and m in (1.7) are given by whereC is a positive constant depending on λ 1 , n and R 2 /R 3 .
From Theorem 1.3, we immediately conclude that if (u, p) ∈ (H 1 loc (Ω)) n+1 satisfies (1.1) and for any N ∈ N, there exists C N > 0 such that |x|<r |u| 2 dx ≤ C N r N , then u vanishes identically in Ω. Consequently, p is a constant in Ω. This is a new strong unique continuation result for the Stokes system with singular coefficients.
By three-ball inequalities (1.6), one can also study the minimal decaying rate of any nontrivial velocity u to (1.1) with a suitable assumption on the coefficients A and B (see [3] for a related result for the Schrödinger equation). Consider (u, p) satisfying (1.1) with Ω = R n , n = 2, 3. Assume here that Then we can prove that Theorem 1.5. Let (u, p) ∈ (H 1 loc (R n )) n+1 be a nontrivial solution to (1.1). Assume that (1.8) holds. Then for any r < 1, there exists c > 0 such that , where c depends on λ 2 , n, |x|<r |u| 2 dx and ζ = 1 + 2C log(1/r) withC given in Remark 1.4.
We can apply Theorem 1.5 to the stationary Navier-Stokes equation.
This paper is organized as follows. In Section 2, we derive suitable Carlemantype estimates. A technical interior estimate is proved in Section 3. Section 4 is devoted to the proofs of Theorem 1.1, 1.3. The proof of Theorem 1.5 is given in Section 5.
2. Carleman estimates. Similar to the arguments used in [11], we introduce polar coordinates in R n \{0} by setting x = rω, with r = |x|, ω = (ω 1 , · · · , ω n ) ∈ S n−1 . Furthermore, using new coordinate t = log r, we can see that where Ω j is a vector field in S n−1 . We could check that the vector fields Ω j satisfy n j=1 ω j Ω j = 0 and n j=1 Ω j ω j = n − 1.
Since r → 0 iff t → −∞, we are mainly interested in values of t near −∞.
It is easy to see that and, therefore, the Laplacian becomes where ∆ ω = Σ n j=1 Ω 2 j denotes the Laplace-Beltrami operator on S n−1 . We recall that the eigenvalues of −∆ ω are k(k + n − 2), k ∈ N, and the corresponding eigenspaces are E k , where E k is the space of spherical harmonics of degree k. It follows that then Λ is an elliptic first-order positive pseudodifferential operator in L 2 (S n−1 ). The eigenvalues of Λ are k + n−2 2 and the corresponding eigenspaces are E k . Denote Then it follows from (2.1) that Motivated by the ideas in [19], we will derive Carleman-type estimates with Note that ϕ β is less singular than |x| −β , For simplicity, we denote ψ(t) = t + log t 2 , i.e.,ψ(x) = ψ(log |x|). From now on, the notation X Y or X Y means that X ≤ CY or X ≥ CY with some constant C depending only on n.
Lemma 2.1. There exist a sufficiently small r 0 > 0 depending on n and a sufficiently large β 0 > 1 depending on n such that for all u ∈ U r0 and β ≥ β 0 , we have that . Using polar coordinates as described above, we have If we set u = e βψ(t) v and use (2.1), then whereC 1 is a positive constant depending on n. From (2.6), using the integration by parts, for t < t 0 and β > β 0 , where t 0 < −1 and β 0 > 0 depend on n, we have that In view of (2.8), using (2.2),(2.3), we see that Substituting (2.9) into (2.8) yields For k such that k(k + n − 2) < 2β 2 , we have On the other hand, if 2β 2 < k(k +n−2), then, by taking t even smaller, if necessary, we get that Finally, using formula (2.3) and estimates (2.11), (2.12) in (2.10), we immediately obtain (2.7) and the proof of the lemma is complete.

2
To handle the auxiliary equation corresponding to the vorticity q, we need another Carleman estimate. The derivation here follows the line in [20].
There exists a sufficiently small number t 0 < 0 depending on n such that for all u ∈ V t0 , β > 1, we have that . If we set u = e βψ(t) v, then simple integration by parts implies By the definition of Λ, we have where, as before, v k is the projection of v on E k . Note that Considering β > (1/2)k and β ≤ (1/2)k, we can get that (2.14) The estimate (2.13) then follows from (2.3).

2
Next we need a technical lemma. We then use this lemma to derive another Carleman estimate. Lemma 2.3. There exists a sufficiently small number t 1 < −2 depending on n such that for all u ∈ V t1 , g = (g 0 , g 1 , · · · , g n ) ∈ (V t1 ) n+1 and β > 0, we have that Proof.. This lemma can be proved by exactly the same arguments used in Lemma 2.2 of [20]. So we omit the proof here.
2 Lemma 2.4. There exist a sufficiently small number r 1 > 0 depending on n and a sufficiently large number β 1 > 2 depending on n such that for all w ∈ U r1 and f = (f 1 , · · · , f n ) ∈ (U r1 ) n , β ≥ β 1 , we have that

15)
where U r1 is defined as in Lemma 2.1.
Proof.. Replacing β by β + 1 in (2.15), we see that it suffices to prove Working in polar coordinates and using the relation e 2t ∆ = L + L − , (2.16) is equivalent to Applying Lemma 2.3 to u = L − w and g = ( n j=1 ω j f j , f 1 , · · · , f n ) yields  3. Interior estimates. To establish the three-ball inequality for (1.1), the following interior estimate is useful.
The proof of Theorem 1.1 is complete.
We now turn to the proof of Theorem 1.3. We fix R 2 , R 3 in Theorem 1.1. By dividing |x|<R2 |u| 2 dx on the three-ball inequality (1.5), we have that 1 ≤ C(  We now end the proof of Theorem 1.3.

5.
Proof of Theorem 1.5. We prove Theorem 1.5 in this section. Let us first choose a > max{2,R −1 }, whereR is given in Theorem 1.1. By doing so, we can see that if we set R 2 = ar and R 3 = a 2 r, then R 2 /R 3 <R for r > 0. Now let 0 < r < 1 and define R 2 , R 3 accordingly. Let |x| = t. We pick a sequence of points 0 = x 0 , x 1 , · · · , x N =x such that |x j+1 − x j | ≤ r. We shall prove the desired estimate iteratively. To see how the iteration goes, let us assume that |x−x l |<r |u| 2 dx ≥ r m l for some m l > 0 since u is nontrivial. By Theorem 1.3 and Remark 1.4, we have that |x−x l+1 |<r Using the boundedness assumption of u (see (1.8)) and r < 1, we can deduce that |x−x l+1 |<R3 |u| 2 dx |x−x l+1 |<R2 |u| 2 dx ≤ a 2n λ 2 2 r n−m l ≤ r −s−m l (5.2) for some s depending on λ 2 and n. Note that we can assume s ≤ m l by choosing a larger m l . It follows from (5.2) that r m ≥ rC (s+m l ) log(1/r) ≥ r 2m lC log(1/r) .