We study the unique continuation properties of solutions of the Navier-Stokes equations. We take advantage of rotation transformation of the Navier-Stokes equations to prove the “logarithmic convexity” of certain quantities, which measure the suitable Gaussian decay at infinity to obtain the Gaussian decay weighted estimates, as well as Carleman inequality. As a consequence we obtain sufficient conditions on the behavior of the solution at two different times and which guarantee the “global” unique continuation of solutions for the Navier-Stokes equations.

1. Introduction

In this paper, we study the unique continuity of the Navier-Stokes equations:For (1), the existence of the Leray solutions [1] can be found in [24] (see also [59] for the existence of the general solutions). The regularity of (1) () is an open problem, but some results of the partial regularity can refer to [1012].

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 , and obtained that the strong solutions to (1) which decay in at the rate of have the following point-wise space-time decay, for ,with , , and . The decay for solutions of (1) was studied in [1417]. 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 -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 so-called 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 obtained. In [28], interpolation arguments and Sobolev imbedding theorem led to an Carleman estimate therefore to a unique continuation theorem. In [2932], the “global” unique continuation for the Schrödinger equations was discussed.

For the stationary Navier-Stokes equations:in three dimensions, Finn [33] showed that and ; then, is trivial (). Later Dyer and Edmunds [34] proved that if is bounded, and for all , then is trivial (see also [35]). In [36], Lin et al. showed that, for , if is bounded in , then any nontrivial of (3) cannot decay faster than certain double exponential at infinity (see Corollary 1.6 [36]). In [37], they improved on this result in [36] and studied the asymptotic behavior of solutions of the stationary Navier-Stokes equations in an exterior domain, through assuming , , or and choosing the appropriate Carleman estimates based on Lemma 2.4 in [38], combining interior estimates, showed that any nontrivial solution obeyed a minimal decaying rate at infinity. Our goal is to obtain sufficient conditions on the behavior of the solution at two different times and which guarantee that the solution of (1) is trivial.

For the heat equation, applying Hardy’s uncertainty principle [39] to , and would be in , and implies . Then, backward uniqueness arguments [40] show that . Moreover, due to the result in [31]: If , , and , then . This result can be rewritten in terms of the free solution of the Schrödinger equation:That means if , , then when , . By applying “logarithmically convex” and Carleman estimates of equations after the Appel transformation, Escauriaza et al. [31] showed the following.

(1) If and satisfieswhere are positive, , and , are finite,then .

(2) If satisfieswhere is bounded in and , , are in , then .

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).

Theorem 1. If satisfies (1) and there are constants which satisfy the following inequalities:We also assume that and are in . Then for .

Our arrangement is as follows. In Section 2, we introduce variable transformation of the ; thus, we can reduce the information of the tension item and simplify the equations. Hence we get the equations of the tensor . 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 estimates of . But in this Gaussian weight 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 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 . 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 .

We give the notations in this paper.

Let be 2-order tensor, and define

, are the symmetric and skew-symmetric operators, shows the partial derivative on about coefficients of the operator , and commutator .

means with compact support.

2. Reduced System

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

In detail, we also introducewhere denotes the transpose of . Then it is easy to prove

In fact,Noticing that , (12) becomes .

Moreover,Applying on the equations , we havewhere . In fact, by the definitions of and , we getThereforeBy , we easily get (15).

For , due to , it is easy to see that . Hence, for , we only need to consider the system:for , we only need to consider the equations:

In order to get the unique continuation of (1), putting together (18) and (19), it suffices to considerwhere , .

3. Gaussian Weighted Estimates

In this section, we consider Gaussian weighted estimates of and in (20); that is,

Proposition 2. If  , ,     satisfies, and , thenwhere .

Proof. Let , ; then,Denote by . Without loss of generality, assume that . Let such thatwithLet , so that nondecreasing withLet so that and as .
Taking , andUsing thatSo, one gets the equations of ,Multiplying the both sides of (30) by and integrating over , we obtainUsing the integration by parts, it follows thatUsing the Young inequality, we haveDenote byIntegrating (33) over , we obtainLetting , we obtain that The above variables , are changed to , ; we getOn the other hand, it is easy to prove thatIn fact, noticing thatand usingwe haveThat is, (38) holds.
Using (37) and (38), we haveIn (42), taking , it follows thatReplacing by in (43) yieldsThat is, (23) holds. The proof of Proposition 2 is completed.

Remark 3. Taking the gradient operator to the both sides of (21), we get the systems about ,where replace , by , . Using Proposition 2, we get the estimates of the ,where , .

Proposition 2 shows that relates to the initial value of , and Remark 3 means that relates to the initial value of and . But the key of this paper is to control the behavior of solution in interval by solution at two different times . The following Lemma introduces an abstract result (for the tensor ) which shows how to get the “logarithmic convexity” property, which is analogous to Lemma 2 in [31] (for the complex function).

Lemma 4. Let be a symmetric operator and a skew-symmetric operator, both allowed to depend on the time variable. is a positive function; is a reasonable real -order tensor function.
DenoteMoreover, if there exist constants , , and such thatare achieved, then is “logarithmically convex” in and there is a universal constant such that

Proof. On one hand,Because is a skew-symmetric operator and , thusOn the other handDifferentiating , we getNoticing thatwe haveFrom (52) and (55), it follows thatFrom (55) and (51), it follows thatUsing the Cauchy-Schwarz inequality, we haveMeantime, , soFrom (48) and the inequality above,We remark (52) shows thatwhere is a function andAll together, when ,Therefore, when ,On one hand, the integration of the inequality above over the intervals and shows thatOn the other hand,Combining with (64), it implies (49). This completes the proof of Lemma 4.

Proposition 5. Assume that verifies (21) and are finite as , and letThen, is “logarithmical convexity” in and there is a universal constant such thatwhen .

Proof. Let ; then, from the equation about , we get the equationwhere is a symmetric operator and is a skew-symmetric operator. At this time,Let be the th element of and the partial derivative on . A calculation shows thatThus, if we take ,A formal integration by parts shows thatUsing the two results above and noticing that and , , we get thatWhen , we haveSo, . On the other hand, we getTherefore when , from Lemma 4, we have the “logarithmic convexity” of 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 is finite; then,where , , and are as in Proposition 5.

Proof. Let , from Proposition 5; we havewhere , , and .
Thus, integration over to timing of the formula (56) in Lemma 4 shows thatOn the other hand, integrating the above equality by parts showsThereforethat is,From (78) and (46), we obtainTo be simplified,