Abstract
Consider the time-periodic viscous incompressible fluid flow past a body with non-zero velocity at infinity. This article gives sufficient conditions such that weak solutions to this problem are smooth. Since time-periodic solutions do not have finite kinetic energy in general, the well-known regularity results for weak solutions to the corresponding initial-value problem cannot be transferred directly. The established regularity criterion demands a certain integrability of the purely periodic part of the velocity field or its gradient, but it does not concern the time mean of these quantities.
Keywords:
time-periodic solutions; weak solutions; exterior domain; regularity criterion; Serrin condition; Oseen problem MSC:
35B10; 35B65; 35Q30; 76D03; 76D05; 76D07
1. Introduction
We consider the time-periodic flow of a viscous incompressible fluid past a three-dimensional body that translates with constant non-zero velocity . We assume to be directed along the -axis such that with . In a frame attached to the body, the fluid motion is then governed by the Navier–Stokes equations:
where is the exterior domain occupied by the fluid.
The functions and are velocity and pressure of the fluid flow, is an external body force, and denotes the velocity field at the boundary. The time axis is given by the torus group , which ensures that all functions appearing in Equation (1) are time periodic with a prescribed period .
In this article, we study weak solutions to the problem in Equation (1), and we provide sufficient conditions such that these weak solutions possess more regularity and are actually smooth solutions. In the context of the initial-value problem for the Navier–Stokes equations, these criteria have been studied extensively. Existence of weak solutions was shown several decades ago in the seminal works by Leray [] and Hopf [] together with a corresponding energy inequality, but it remained unclear for many decades whether solutions in this Leray–Hopf class are unique, even when the external forcing is smooth (or even 0). Note that Albritton, Brué and Colombo [] recently showed that there are forcing terms such that multiple Leray–Hopf solutions to the initial-value problem exist, so that uniqueness fails for general forcing terms. However, Leray–Hopf solutions come along with a weak-strong uniqueness principle that states that weak solutions coincide with strong solutions if the latter exist. This also motivated the development of criteria that ensured higher regularity of weak solutions. The first results in this direction are due to Leray [] and Serrin [], who showed that, if a weak solution is an element of for some such that , then it is a strong solution and smooth with respect to the spatial variables. Since then, there appeared many other regularity criteria that ensured higher-order regularity of a weak solution to the initial-value problem; see [,,,,,] and the references therein.
To obtain similar regularity results for weak solutions to the time-periodic problem in Equation (1), the first idea might be to identify these with weak solutions to the initial-value problem for a suitable initial value. However, this procedure is not successful in the considered framework of an exterior domain since regularity of weak solutions to the initial-value problem is usually investigated within the class , but weak solutions u to the time-periodic problem are merely elements of at the outset; see Definition 1 below. To see that we cannot expect the same integrability as for the initial-value problem, observe that every weak solution to the steady-state problem is also a time-periodic solution. In general, these steady-state solutions do not have finite kinetic energy but only belong to for ; see Theorem 4 below. Therefore, one cannot reduce the time-periodic situation to that of the initial-value problem.
For the formulation of suitable regularity criteria for time-periodic weak solutions, we decompose functions into a time-independent part, given by the time mean over one period, and a time-periodic remainder part. To this decomposition, we associate a pair of complementary projections and such that
Then, is called the steady-state part of u, and denotes the purely periodic part of u.
In this article, we consider weak solutions to (1) in the following sense.
Definition 1.
Let and . A function is called weak solution to (1) if it satisfies the following properties:
- i.
- , , in , on ,
- ii.
- ,
- iii.
- the identity
The existence of weak solutions in the sense of Definition 1 satisfying an associated energy inequality was shown in [] for . Their asymptotic properties as were investigated in [,,]. For these results, it was necessary to ensure higher regularity of the solution u, which was achieved by assuming that
holds for some . Moreover, it was shown in [] that u satisfies an energy equality if Equation (2) holds for some and with . It is remarkable that, in both cases, the additional integrability is only assumed for the purely periodic part , but not for the whole weak solution u as is achieved for the initial-value problem. The main result of this article is in the same spirit and can be seen as an extension of the regularity results used in [,,]. More precisely, we consider the criteria
If the domain has a smooth boundary and the data are smooth, then both lead to smooth solutions.
Theorem 1.
Let be an exterior domain with a boundary of class , and let . Let and , and let u be a weak time-periodic solution to (1) in the sense of Definition 1 such that (3) or (4) is satisfied. Then, there exists a corresponding pressure field such that is a smooth solution to Equation (1) and
As an intermediate step, we show the following result that assumes less smooth data.
Theorem 2.
Let be an exterior domain with boundary of class , and let . Let f and be such that
Let u be a weak time-periodic solution to Equation (1) in the sense of Definition 1 such that Equation (3) or Equation (4) is satisfied. Then, and satisfies
and there exists a pressure field with and such that
and the identities in Equation (1) are satisfied in the strong sense.
Additionally, if Ω has a -boundary, and if and for some , then
Theorems 1 and 2 are the main results of this article and will be proved in Section 5.
Comparing the regularity criteria of Theorems 1 and 2 with those used in [,,], we see that the present article extends them in two directions. Firstly, by Equation (3), we extend the range of admissible parameters , in the sufficient condition (2) by also allowing the mixed case . Secondly, Equation (4) is an alternative condition on certain integrability of the purely periodic part of the gradient . In particular, we can replace the assumption in Equation (2) for some with one of the assumptions in Equation (3) or Equation (4) in the main results of [,,], and the results on the spatially asymptotic behavior of the velocity and the vorticity field derived there are also valid under the alternative regularity criteria from Equation (3) or Equation (4).
In Section 2, we next introduce the general notation used in this article. In Section 3, we recall the notion of Fourier multipliers in spaces with mixed Lebesgue norms and introduce a corresponding transference principle, from which we derive an embedding theorem. Section 4 recalls a well-known regularity result for the steady-state Navier–Stokes equations, and it contains a similar result for the time-periodic Oseen problem, which is a linearized version of Equation (1). Finally, Theorems 1 and 2 will be proved in Section 5, and we conclude the paper by a short outlook in Section 6.
2. Notation
For the whole article, the time period is a fixed constant, and denotes the corresponding torus group, which serves as the time axis. The spatial domain is usually given by a three-dimensional exterior domain , that is, the domain is the complement of a compact connected set. We write and for partial derivatives with respect to time and space, and we set and , where we used Einstein’s summation convention.
We equip the compact abelian group with the normalized Lebesgue measure given by
and the group , which can be identified with the dual group of , with the counting measure. The Fourier transform on the locally compact group , , and its inverse are formally given by
where the Lebesgue measure is normalized appropriately such that defines an isomorphism with inverse . Here, is the so-called Schwartz–Bruhat space, which is a generalization of the classical Schwartz space in the Euclidean setting; see [,]. By duality, this induces an isomorphism of the dual spaces and , the corresponding spaces of tempered distributions.
By and as well as and , we denote the classical Lebesgue and Sobolev spaces, and and denote the respective classes of locally integrable functions. We define homogeneous Sobolev spaces by
where denotes the collection of all (spatial) weak derivatives of the u of m-th order. We further set
where is the class of compactly supported smooth functions on . For and a (semi-)normed vector space X, denotes the corresponding Bochner–Lebesgue space on , and
The projections
decompose into a time-independent steady-state part and a purely periodic part .
We further study the fractional time derivative for , which is defined by
for . By Plancherel’s theorem (see [][Prop. 3.1.16] for example), one readily verifies the integration-by-parts formula
for all . By duality, extends to an operator on the distributions . Note that in general we have for , but
holds for and . If for some , we usually write .
3. Transference Principle and Embedding Theorem
To analyze mapping properties of the fractional derivative and other operators, we need the notion of Fourier multipliers on the locally compact abelian group for . We are interested in multipliers that induce bounded operators between mixed-norm spaces of the form for . We call an -multiplier if there is such that
and we call an -multiplier if there is such that
The smallest such constant C is denoted by and and called the multiplier norm of M and m, respectively. The following transference principle enables us to reduce multipliers on to multipliers on .
Proposition 1.
Let , and let be an -multiplier. Then, is an -multiplier with norm
Proof.
The statement can be shown as in [], where a transference principle from scalar-valued -multipliers to -multipliers was shown. For a more direct and modern approach, one may also follow the proof of ([] [Proposition 5.7.1]), where an operator-valued version of the result from [] was established. □
We now apply this transference principle to show the following result, which is an extension of ([] [Theorem 4.1]) to the case of mixed norms. Moreover, we also take fractional time derivatives into account.
Theorem 3.
Let , , be a bounded or exterior domain with Lipschitz boundary, and let . For , let
and, for , let
Then, there is such that all satisfy the inequality
Proof.
For the proof, we proceed analogously to ([] [Theorem 4.1]). However, we have to modify some arguments in the case , and we also derive estimates for the fractional time derivative, which is why we give some details here. Using Sobolev extension operators and the density properties of , it suffices to show the estimate from Equation (13) for and with .
We begin with the estimate of u. By means of the Fourier transform, we obtain
where
Here, is the delta distribution on , that is, with if and only if . We can extend to a continuous function in a trivial way such that . One readily shows that m satisfies the Lizorkin multiplier theorem ([] [Corollary 1]), so that the function m is an -multiplier. Due to the transference principle from Proposition 1, this implies that M is an -multiplier, and we have
Moreover, from ([] [Example 3.1.19]) and ([] [Proposition 6.1.5]), we conclude
for and . Young’s inequality thus implies that defines a continuous linear operator if , and defines a continuous linear operator . Therefore, Equation (14) yields
Invoking now Equation (15), we arrive at the desired estimate for u if . Since is compact, the estimate for follows immediately.
The remaining estimates of , and can be shown in the same way as those for u. Note that, for the estimates of and , the procedure has to be slightly modified since the trivial extension of the corresponding multipliers to is not continuous. To demonstrate this, we focus on the estimate for , which means nothing else than the boundedness of the linear operator
Similarly to the above, this boundedness follows if the function
is an -multiplier for . Note that its trivial extension is not a continuous function in , which is necessary for application of the transference principle from Proposition 1. However, since , we can introduce a smooth cut-off function with and such that for . We define
Then, m is a smooth function with , and one readily verifies that m satisfies the multiplier theorem by Lizorkin ([] [Corollary 1]). Finally, Proposition 1 shows that M is an -multiplier, which implies the estimate for . □
As mentioned in the proof, the lower bound 1 for and is valid since the torus has finite measure. In the same manner, the lower bound for and can be replaced with 1 if is a bounded domain.
In ([] [Theorem 4.1]), a homogeneous version of Theorem 3 was shown, but only in the case . Modifying the proof in [] and using similar arguments as above, this result is easily extended to the case .
We might also formulate the assumptions on the integrability exponents in Theorem 3 as follows: Let such that
where in each of the four conditions the left < can be replaced with ≤ if the respective lower bound is different from 0.
4. Preliminary Regularity Results
As a preparation for the proof of the main theorems, we first consider the steady-state Navier–Stokes equations
and recall the following result on the regularity of weak solutions.
Theorem 4.
Let be an exterior domain with a -boundary. Let such that
for and for . If v is a weak solution to Equation (16), then there exists an associated pressure field p such that
and Equation (16) is satisfied in the strong sense. Additionally, if there exists such that Equation (17) holds for all , then v satisfies Equations (7) and (8). Moreover, if Ω has -boundary and and , then Equation (6) holds and for all .
Proof.
See ([] [Lemma X.6.1 and Theorem X.6.4]). □
We further derive a similar regularity result for weak solutions to the time-periodic Oseen problem, which is the linearization of Equation (1) given by
Here, we focus on the case of purely oscillatory data. To shorten the notation, we denote the mixed-norm parabolic space by
Lemma 1.
Let be an exterior domain of class , and let be as in Equation (5b), and let for some such that and . Let with and be a weak solution to Equation (18), that is, on , and
for all . Then, , and there exists such that is a strong solution to Equation (18).
Proof.
For , the result was shown in ([] [Lemma 5.1]). Arguing in the same way, we can show that it suffices to treat the case . In this case, first consider a solution of the time-periodic Stokes problem, that is, the system
We now use the result from ([] [Theorem 5.5]) on maximal regularity for this system for right-hand sides in . From this, we conclude the existence of a unique solution with and . The embedding Theorem 3 implies that for with . We again employ the the maximal regularity result from ([] [Theorem 5.5]) to obtain the existence of a unique solution to
such that and for all as above. Employing Theorem 3 once more, we see that for any . In particular, we can choose , that is, we have . Now we can use the maximal regularity result ([] [Theorem 5.1]) for the Oseen system for the right-hand sides in to find a solution to
such that . Theorem 3 further implies for such that by ([] [Theorem 5.5]). Repeating this argument once again, we obtain for . In total, we see that and satisfy the Oseen system from Equation (18) and . To conclude that , one can now proceed as in the proof of ([] [Lemma 5.1]). The regularity of the pressure follows immediately. □
Observe that, for the proof, we combined two results on maximal regularity: one for the Stokes problem in Equation (20) for right-hand sides in , and one for the Oseen problem in Equation (18) for right-hand sides in . The argument could be shortened severely if such a result would be available for the Oseen problem in Equation (18) for right-hand sides in . For a proof, one can use the approach developed in ([] [Theorem 5.5]), which would also give corresponding a priori estimates.
5. Regularity of Time-Periodic Weak Solutions
Now, we begin with the proof of Theorem 2, for which we proceed by a bootstrap argument that increases the range of admissible integrability exponents step by step. To shorten the notation, we introduce the Serrin number
For the whole section, let f and satisfy Equation (5), and let u be a weak solution in the sense of Definition 1. We decompose u and set and .
We first show that the definition of weak solutions already implies some degree of increased regularity and that there exists a pressure such that the Navier–Stokes equations are satisfied in the strong sense.
Lemma 2.
There exists a pressure field such that
and the Navier–Stokes equations in Equation (1) are satisfied in the strong sense. More precisely, it holds that
and
Proof.
From the integrability of w, we conclude with Hölder’s inequality that . We thus have , and Theorem 4 yields the existence of p such that Equation (24) and Equations (21) and (22) also hold.
To obtain the regularity statement for w, note that Equation (22) implies for all . Moreover, we have for all and with by the Sobolev inequality and interpolation. By virtue of Equation (22) and Hölder’s inequality, we conclude that for all and with . In consequence, we obtain
for all such q and r. Now, Lemma 1 yields the existence of a pressure such that Equation (25) is satisfied in the strong sense and Equation (23) holds. □
In the following lemmas, we always assume that for some given , and the goal is to extend the range of one of the parameters q or r while the other one remains fixed. We use the assumption on additional regularity from Equation (3) or Equation (4), or the embedding properties from Theorem 3 to conclude that
for a class of parameters , and
for a class of parameters , and we use Lemma 2 or Theorem 4 to deduce
for certain and
for certain . Then, Hölder’s inequality yields suitable estimates of the nonlinear terms and of the total right-hand side
for a certain class of parameters . Invoking now the regularity result from Lemma 1, we conclude .
As a preparation, we first derive suitable estimates of the nonlinear terms if we have . In the next lemma, we start with the nonlinear term
and we show better integrability for v and for sufficiently large q.
Lemma 3.
- i.
- and , and
- ii.
- and .
Moreover, if or , then the steady-state part v satisfies Equations (7) and (8).
Proof.
At first, Theorem 3 yields Equation (26) for and , and Equation (27) for and , so that we deduce Equation (31) for and as asserted in i. Moreover, Theorem 3 yields for as well as for . Now, Hölder’s inequality implies the integrability of asserted in ii.
If, additionally, , then the lower bound in i. is smaller than 1, so that for some . The same follows from ii. for . Now, Theorem 4 yields Equations (7) and (8). □
Next, we treat the nonlinear terms that involve v and , namely we show that implies
and
for suitable parameters .
Lemma 4.
- i.
- and , and
- ii.
- and ,
- iii.
- and , and
- iv.
- and if , and
- v.
- and .
Proof.
Theorem 3 implies Equation (27) for and as well as for and for small. Moreover, by Lemma 2, we have Equation (28) for , and Hölder’s inequality implies Equation (32) for and as in i. or ii.
Theorem 4 and Lemma 2 yield Equation (29) for all . Theorem 3 implies Equation (26) for and . Hölder’s inequality now yields Equation (33) for and as in iii. Additionally, if , then we obtain Equation (7) by Lemma 3. Theorem 3 further implies Equation (26) for and , so that Hölder’s inequality yields Equation (33) for and as in iv.
For v., we distinguish two cases. Firstly, if , then Theorem 3 implies Equation (26) for and for , and Lemma 2 yields Equation (29) for all , so that Hölder’s inequality implies Equation (33) for and . Secondly, if , then we use Lemma 3 again to conclude Equation (7). Moreover, Theorem 3 yields Equation (26) for and , and we conclude Equation (33) for and . Combining both cases, we obtain v. □
The results from Lemmas 3 and 4 are not sufficient to conclude the proof, and we need to invoke the additional regularity assumptions from Equation (3) or Equation (4) to obtain Equation (31) for other parameters and . We define by
Lemma 5.
- i.
- and , and
- ii.
- and .
Proof.
At first, let us assume Equation (3). Then, and Theorem 3 with yields Equation (27) for and . Combining this with Equation (3) and using Hölder’s inequality, we obtain Equation (31) for q, r as in i. Moreover, we have , and Theorem 3 with yields Equation (27) for and . Combining this with Equation (3) and using Hölder’s inequality, we obtain Equation (31) for q, r as in ii.
Now, let us assume Equation (4). From Theorem 3 with , we deduce Equation (26) for and . Combining this with Equation (4) and using Hölder’s inequality, we also obtain Equation (31) in this case for q, r as in i. Moreover, we have , and Theorem 3 with yields Equation (26) for and . Combining this with Equation (4) and using Hölder’s inequality, we also obtain Equation (31) in this case for q, r as claimed in ii. □
Now, we have prepared everything to iteratively increase the range of parameters q, r such that . By Lemma 2, we start with q, r such that . In particular, both parameters cannot be chosen as large, and we use Lemmas 4 and 5 to extend the range of admissible parameters. An iteration leads to sufficiently large parameters such that Lemma 3 can be invoked to further iterate until the full range is admissible for both parameters, which proves the regularity result from Theorem 2.
Proof of Theorem 2.
As a first step, we show that for all and all . To do so, observe that both Equations (3) and (4) imply that or . In what follows, we distinguish these two cases:
Consider the case at first. We show that for all and . Let and with , so that by Lemma 2. Then, we have Equation (31) for , as in i. of Lemma 5, we have Equation (32) for , as in i. of Lemma 4, and we have Equation (33) for , as in iii. of Lemma 4. We thus obtain Equation (30) for and . Since , this interval is non-empty, and by the regularity result from Lemma 1, we conclude for . Repeating this argument iteratively with q replaced with a suitable within this range, we obtain for . If , this completes the first step. If this is not the case, we repeat the above argument for , but we use iv. of Lemma 4 instead of iii., which leads to Equation (30) for and , and thus for for . Repeating now this argument a sufficient number of times for admissible instead of q, we finally arrive at for all and .
Since we assume , we can now choose . Let such that . The previous step implies , and we conclude Equation (31) for , as in ii. of Lemma 3, we have Equation (32) for , as in ii. of Lemma 4, and we have Equation (33) for , as in v. of Lemma 4. We thus obtain Equation (30) for and . Invoking Lemma 1, we obtain for , and an iteration as above yields for all and all .
Now, let and . Then, and, since , we have Equation (31) for , as in i. of Lemma 3, we have Equation (32) for , as in i. of Lemma 4, and we have Equation (33) for , as in iv. of Lemma 4. We thus obtain Equation (30) for and , and Lemma 1 yields for . An iteration of this argument leads to for all and all .
Now consider the case . We first extend the range for r and show that for all and . For this, fix . Lemma 2 yields for some such that . Then, we have Equation (31) for , as in ii. of Lemma 5, we have Equation (32) for , as in ii. of Lemma 4, and we have Equation (33) for , as in v. of Lemma 4. We thus obtain Equation (30) for and . Using the regularity result from Lemma 1, we now obtain for . Repeating this argument with r replaced with some in this range, we can successively increase the admissible range for until we obtain for all and .
Since , we can choose , and from for and we conclude Equation (31) for , as in i. of Lemma 3, we have Equation (32) for , as in i. of Lemma 4, and we have Equation (33) for , as in iii. of Lemma 4. We thus obtain Equation (30) for and . Invoking Lemma 1, we obtain for , and an iteration as above yields for all . Now, we can choose , and repeating the argument with iv. of Lemma 4 instead of iii., we obtain for . Another iteration now leads to for all and all if .
Now, let and . Then, and, since , we have Equation (31) for , as in ii. of Lemma 3, we have Equation (32) for , as in ii. of Lemma 4, and we have Equation (33) for , as in v. of Lemma 4. We thus obtain Equation (30) for and . Invoking Lemma 1, we obtain for , and an iteration as above yields for all and all .
Combining these two cases and using that is compact, we have shown that for all and . In particular, v satisfies Equations (8) and (7) by Lemma 3, and we have Equation (6) by Lemma 2. To conclude Equation (9), note that Theorem 3 implies Equations (26) and (27) for and , so that Equation (31) holds for and , and i. and iv. of Lemma 4 yield Equation (32) for and , and Equation (33) for and . We thus have obtained Equation (30) for and , and, from Lemma 1, we conclude for all and . Repeating the argument once more, we finally obtain Equation (9). Moreover, Equation (10) is a direct consequence of Equations (6)–(9) by virtue of Equations (24) and (25).
Finally, Equation (11) follows from Theorem 4 and the additional assumptions on , f and since Equation (9) implies that for any . □
Proof of Theorem 1.
At first, we increase the time regularity of the solution inductively in steps of half a derivative. For , let and . We show that, for every , we have
By Theorem 2, there exists a pressure field such that is a strong solution to Equation (1) with the regularity stated in Equations (6)–(11). In particular, this shows Equation (34) for . Now, assume that has the asserted regularity stated in Equation (34) for all . Then, Theorem 3 implies
for . Let and multiply Equation (1a) by . Since for , after integrating by parts in space and time as well as by means of Equation (12), we obtain
where
In virtue of the smoothness of the boundary data and the regularity of , we see that is a weak solution to the Navier–Stokes equations from Equation (1) for the right-hand side , which is an element of for all . For the first two terms in the definition of , this follows from the assumptions and from Equation (35). For the term , we distinguish two cases.
If is an odd number, then this term is an element of if and only if is an element of for . We write
We can estimate the terms of this sum as
for , where the respective right-hand side is finite due to Equations (7)–(9) and the embedding Theorem 3 as well as Equation (35) for . If is an even number, then if and only if this is true for
By Theorem 3, this is the case if for . For example, for the terms with derivatives of highest order we obtain
which are all finite by the same argument as above. Similarly, this follows for the lower-order terms.
In summary, we obtain for all in both cases. By Equation (35), the function is subject to both regularity assumptions from Equations (3) and (4), and Theorem 2 implies that satisfies Equation (34) for . We thus have shown Equation (34) for all .
To increase the spatial regularity, we recall that is a strong solution by Theorem 2, so that the N-th time derivative, , satisfies the Stokes system
a.e. in , where
Since for , Theorem 2 and Equation (34) imply for all and all such that , where we define , and is the ball with radius R and centered at . By a classical regularity result for the steady-state Stokes problem (see [] [Theorem IV.5.1] for example), we obtain for all sufficiently large and all . This implies , and can again apply ([] [Theorem IV.5.1]) to deduce . Iterating this argument, we finally obtain
for all , all and all such that . This completes the proof. □
6. Conclusions and Outlook
As the main result, this paper contains new regularity criteria for time-periodic weak solutions to the Navier–Stokes equations with a non-zero drift term in exterior domains. These criteria are given in the form of a Serrin-type condition on the purely periodic part of the velocity field u or its gradient, but they do not involve the steady-state part . This is a severe difference to known regularity results for the initial-value problem. Moreover, this article generalizes the regularity criterion used in [,,], so that the results on the asymptotic behavior of time-periodic solutions also hold under more general assumptions.
A natural question for further research would be whether the conditions Equations (3) and (4) can be extended to the critical case, that is, if one can still obtain smoothness of weak solutions if the strict inequalities in Equations (3) and (4) are replaced with equalities. In this case, the presented proof is not applicable, but analogous results are well known for the initial-value problem.
Moreover, the present article focuses on the case , that is, the flow around a translating body. In the case , corresponding to the flow around a body at rest, the above bootstrap argument cannot be employed since the decay properties of the velocity field are worse, similarly to the properties of the time-independent problem (see [] [Ch. X]). Moreover, as mentioned above, time-periodic solutions cannot be expected to have finite kinetics and thus cannot be identified with Leray–Hopf weak solutions to the initial-value problem. Therefore, it remains an open question how to establish regularity criteria such as Equation (3) or Equation (4) in the case .
Funding
This research received no external funding.
Data Availability Statement
Not applicable.
Conflicts of Interest
The author declares no conflict of interest.
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