On Unique Continuation for Navier-Stokes Equations

and Applied Analysis 3 󵄩󵄩󵄩󵄩 f (x, t) 󵄩󵄩󵄩󵄩L2(Rn×[0,1]) = (∫ 1 0 󵄩󵄩󵄩󵄩 f (⋅, t) 󵄩󵄩󵄩󵄩 2 dt) 1/2

Due to the fact that our consequence needs some asymptotic behavior of the solutions to (1) as conditions, we first mention some the space-time asymptotic behavior of the solutions.In [13], Amrouch et al. studied the space-time asymptotic behavior of the solutions, and their derivatives, to (1) in dimension 2 ≤  ≤ 5, and obtained that the strong solutions to (1) which decay in  2 at the rate of ‖()‖ 2 ≤ ( + 1) − have the following point-wise space-time decay, for 0 ≤  ≤ /2,        (, )     ≤  , 1 with  0 = (1 − 2/)(/2 +  + /4), || = , and  > (/4).The  2 decay for solutions of (1) was studied in [14][15][16][17].On other hand, the backward uniqueness for parabolic equations will be used in our paper.In [18], Escauriaza et al. proved a backward uniqueness result for the heat operator with variable lower order terms, which implies the full regularity of  3,∞ -solutions of the three-dimensional Navier-Stokes equations.Some backward results of the Navier-Stokes equations can refer to [19,20].The unique continuation is best understood for second order elliptic operators, in which the powerful technique socalled Carleman weighted estimate played a central role (see [21,Chapter 17] and [22]).In [21,Chapter 28], the Calderón uniqueness theorems for some general linear partial differential operators were obtained by Carleman estimates.The proofs in Chapter 28 of [21] relied on factorization in first order pseudodifferential operators.A careful study of these factors led to more general forms of the Calderón uniqueness theorem.In [23], Saut and Scheurer proved a unique continuation theorem when  was a second order parabolic equation in the first section.Their proof is simple and based on the derivation of a Carleman estimate which is reminiscent of the classical Carleman estimates for second order elliptic operators.This Carleman inequality allows the weakening of the smoothness assumptions on the principal operator .And they extended also these results to some mixed parabolic-elliptic systems and some higher order parabolic equations.For the parabolic equations [24], the stokes equations [25], and the Navier-Stokes equations [26,27], the similar "local (space)" uniqueness results were 2 Abstract and Applied Analysis obtained.In [28], interpolation arguments and Sobolev imbedding theorem led to an   ( > 2) Carleman estimate therefore to a unique continuation theorem.In [29][30][31][32], the "global" unique continuation for the Schrödinger equations was discussed.
(1) If  ∈ ([0, 1],  2 (R  )) and satisfies where ,  are positive,  < 2, and then  ≡ 0. ( where (, ) is bounded in R  × [0, 1] and  < 1, ,  || 2 / 2 (1) are in  2 (R  ), then  ≡ 0. Based on these results above, it is natural to expect that Hardy's uncertain principle holds on Navier-Stokes equations (1).In this paper, our aim is to prove the the following unique continuation theorem of Navier-Stokes equations (1).(1) and there are constants  0 ,  1 ,  2 which satisfy the following inequalities: We also assume that  || 2 curl (0) and Our arrangement is as follows.In Section 2, we introduce variable transformation of the curl; thus, we can reduce the information of the tension item and simplify the equations.Hence we get the equations of the tensor  = curl .In Section 3, using the transformation of weighted function and constructing "logarithmic convexity" of the solutions of the equations about , we get the Gaussian weight  2 estimates of .But in this Gaussian weight  2 estimates, we need to justify the validity of the arguments in Proposition 5. Due to the fact that the equations of  include the term ( ⋅ ∇), we cannot use the cut-off method as in [31] to justify the validity.We thus first give a Gaussian weight  2 preestimates (see Proposition 2).In Section 4, we prove a Carleman estimate about .By the Carleman estimate above, we mainly stress the unique continuation for the equations about  in R  × [0, 1].This means we accordingly get the unique continuation for the equation about .According with the conditions that have been given, we lastly analyze the unique continuation for the given Navier-Stokes equations about  in R  × [0, 1].
We give the notations in this paper.

Reduced System
In order to simplify the equations, we introduce , which is the curl of the solutions of (1): Thus we transform (1) into equations about .

Gaussian Weighted Estimates
In this section, we consider Gaussian weighted estimates of  and ∇ in (20); that is, where  > 0.
Taking   () =  1, ( 1 ) 2, ( 2 ) . . . , (  ), and Using that So, one gets the equations of V  , Multiplying the both sides of ( 30) by V  and integrating over R  , we obtain Using the integration by parts, it follows that Using the Young inequality, we have Denote by Integrating ( 33) over (0, ), we obtain Letting  → ∞, we obtain that The above variables ,  are changed to , ; we get On the other hand, it is easy to prove that In fact, noticing that and using we have That is, (38) holds.
Remark 3. Taking the gradient operator ∇ to the both sides of ( 21), we get the systems about ∇, where replace ,  by ( −1/2 ,  −1 ), ( −1/2 ,  −1 ).Using Proposition 2, we get the estimates of the ∇, where  > 0, ( 0 , ) = (14) −1 (3 ∇‖ and ‖ 196|| 2 ‖.But the key of this paper is to control the behavior of solution in interval [0, 1] by solution at two different times  = 0, 1. The following Lemma introduces an abstract result (for the tensor V) which shows how to get the "logarithmic convexity" property, which is analogous to Lemma 2 in [31] (for the complex function).Lemma 4. Let S be a symmetric operator and A a skewsymmetric operator, both allowed to depend on the time variable. is a positive function; V(, ) is a reasonable real 2-order tensor function. Denote Moreover, if there exist constants  0 ,  1 , and are achieved, then log () is "logarithmically convex" in [0, 1] and there is a universal constant  such that Proof.On one hand, Because A is a skew-symmetric operator and (AV, V) = 0, thus (51) On the other hand Differentiating () = (SV, V), we get Noticing that we have From ( 52) and (55), it follows that From ( 55) and (51), it follows that Using the Cauchy-Schwarz inequality, we have From (48) and the inequality above, We remark (52) shows that where ( 1) is a function and All together, when 0 ≤  ≤ 1, Therefore, when 0 ≤  ≤  ≤  ≤ 1,   (log  () +  (1)) ≤   (log  () +  (1)) . ( On one hand, the integration of the inequality above over the intervals 0 ≤  ≤  and  ≤  ≤ 1 shows that Combining with (64), it implies (49).This completes the proof of Lemma 4.
Proof.Let V =   ; then, from the equation about , we get the equation where S = Δ +  2 |∇| 2 +    is a symmetric operator and At this time, Let   be the th element of  and   V the partial derivative on   .A calculation shows that Thus, if we take  = || 2 , A formal integration by parts shows that Using the two results above and noticing that ‖ ⋅ ‖ ∞ ≤  1 and ‖∇‖ ∞ , ‖∇‖ ∞ ≤  0 , we get that When  > 5 0 /16, we have So, S  + [S, A] ≥ − 0 .On the other hand, we get Therefore when V =  || 2 , from Lemma 4, we have the "logarithmic convexity" of () = ‖ || 2 ()‖ 2 and then (68) follows.Proposition 2 shows the validity of the previous arguments.This completes the proof of Proposition 5. Proposition 6. Assume that , , , and  are as in Proposition 5 and  > 0 is finite; then, where ,  0 , and  1 are as in Proposition 5.
Proof.Let V =  || 2 , from Proposition 5; we have where  1 = 8 − (5 0 /2),  2 = 32 3 , and Thus, integration over [0, 1] to (1 − ) timing   () of the formula (56) in Lemma 4 shows that On the other hand, integrating the above equality by parts shows Therefore Abstract and Applied Analysis 9 that is, From ( 78) and ( 46), we obtain To be simplified, Moreover, from V =  || 2 , by Cauchy-Schwarz inequality and integration by parts, it is easy to obtain Furtherly, Thus, the last two formulae give Applying ( 87) to (84), we have Therefore, there is a constant  such that This completes the proof of Proposition 6.

Carleman Estimates
In this section, let  > 5 0 /16 and the assumptions in Propositions 5 and 6 satisfied; we get the following Carleman estimate.

Proposition 7 (Carleman Estimation
where ⃗  1 = (1, 0, . . ., 0), then, there exist  > 0,  > 0, and  > 0 such that Proof.Let V =   ; then, Therefore where Set then So, A calculation shows that where   is the th element of ,    is the partial derivatives on  of   , and   V is the partial derivatives on V about   .That is, Abstract and Applied Analysis 11 So, On one hand, a calculation implies and, from the definition of , we have So, On the other hand, some estimates below hold from the definition of : Integration by parts shows that we thus have Cauchy-Schwarz inequality) .
Thus, with those inequalities above, we have Taking  > (125/128) 4 0 and and letting  be large enough, it follows that Abstract and Applied Analysis 13 Noticing that we have This completes the proof of Proposition 7.
From Lemma 8, we have the following corollary.