In this work we prove the existence and uniqueness of pathwise solutions up to a stopping time to the stochastic Euler equations perturbed by additive and multiplicative Lévy noise in two and three dimensions. The existence of a unique maximal solution is also proved.
The Euler equations are a set of quasilinear partial differential equations which describe the motion of inviscid fluid flow. The mathematical theory for the deterministic Euler equations have been studied by numerous mathematicians in the past several decades ([5,11,13,22,28,45,46], references therein). The global existence and uniqueness of solutions for the Euler equations is still an open problem in three dimensions.
The two-dimensional stochastic Euler equations have been considered by several authors ([2,4,8,10,14,15,24]). The existence of a martingale solution in a bounded domain is proved in [4] and in a smooth subset of is proved in [8]. The stochastic Euler equations with periodic boundary conditions are considered in [10] and the existence of solution on Loeb space with prescribed Wiener process is proved using nonstandard analysis. An existence and uniqueness theorem of strong solution is proved in [2] and [24]. By considering Euler equation as the equation of geodesic on the volume preserving diffeomorphism group, the authors in [14] obtained the existence of global solutions to 2-D stochastic Euler equations. The weak existence of an -regular solution with Dirichlet and periodic boundary conditions on bounded domains is obtained in [15]. The papers [25,33,34] considered the stochastic Euler equations in 3 dimensions with additive Gaussian noise and the paper [20] considered multiplicative Gaussian noise. Navier–Stokes equations with Lévy noise is considered in the papers [7,18,19,36,37,42], for example. We consider the stochastic Euler equations in two and three dimensions perturbed by Lévy noise and prove the local in time existence and uniqueness of strong solutions generalizing the frequency truncation method used in the deterministic context in [17] and [35], and this is a new technique for stochastic quasilinear PDEs developed here. The use of Fourier-harmonic analysis techniques clarify the abstract treatment of the noise covariance structure and other technical calculations found in the related literature. Moreover, to the best of the authors knowledge, this work appears to be the first in establishing a unique solution to the stochastic Euler equations with jump noise.
The construction of the paper is as follows. In Section 2, we formulate the abstract stochastic incompressible, Euler model of fluid dynamics perturbed by additive Lévy noise. By considering a truncated model in the frequency domain, we prove a-priori energy estimates in , for up to a stopping time (Proposition 2.11). The existence and uniqueness of the local in time strong solution (Theorem 3.11 and Theorem 3.14) is obtained in Section 3 by showing that the family of solutions to the truncated incompressible, stochastic Euler equations is Cauchy. The existence of a unique maximal local strong solution (Theorem 3.16) is also proved in this section. In Section 4, we consider the stochastic Euler equations with multiplicative Lévy noise and discuss the existence and uniqueness of the local strong solution (Theorem 4.4).
Stochastic Euler equations
Let be a given filtered probability space. The incompressible stochastic Euler equations (for ) are given by
where is the velocity field is the pressure field and is the external random forcing. The equations are defined on the whole space , with the initial data
satisfying for . Here () is the Hilbertian Sobolev space of order s. Let us define the divergence free space by
In order to eliminate the pressure, we project the equations onto the space of divergence free functions on , by taking the Helmholtz–Hodge orthogonal projection from onto . Let us now express the projection operator in terms of the Riesz transform. Let us define the Riesz transform by
For and , the Fourier transform of the Riesz transform is . Hence, we have (see [9]). The Riesz transform can also be defined in terms of the singular integral operators (Example 2, page 4, [12], Chapter III, [44]). For , set
where is a ball centered at the origin of radius ε and is the volume of unit ball in . For any , one can deduce the Helmholtz–Hodge orthogonal projection operator as
Equivalently, by making use of the Fourier transform, we find
Hence, is an orthogonal projection onto the kernel of the divergence operator so that and is a pseudodifferential operator of order 0 and belongs to the class ([21]). By using the Fourier transform, it can be seen that the operators and commute:
for all and hence . Let us take the Helmholtz–Hodge orthogonal projection on (2.1) to get
Then the stochastic Euler equations perturbed by additive Lévy noise can be written as an Itô stochastic differential equation in after taking as
where for with and Z is a measurable space (where the solution has its paths) such that . In (2.7), Φ is an operator having the properties discussed below and is a cylindrical Wiener process defined on . The operator Φ has the following properties:
The operator , with ,
.
Here denotes the space of all bounded linear operators from to . For an orthonormal basis in , can be written as , where ’s are a sequence of one-dimensional Brownian motions in . Let be a σ-finite Lévy measure on with an associated Poisson random measure . Let be the compensated Poisson random measure, that is, (Theorem 35, Section 4, [39]). The jump coefficient is an -valued function such that , for any . The processes and are mutually independent.
(An example of the operator Φ).
The symbol class , , consists of the -functions on such that for any multi-index α, there exists a constant with
where for . We shall say that is of order m.
A symbol define an operator, denoted by , by the formula (Section 18.5, [21])
Let be a symbol. A pseudodifferential operator with Weyl symbol a is defined by the oscillating integral
An operator with Weyl symbol a is self-adjoint if and only if a is a real function ([21]). We can write as
and the is given by
If ( with ), then extends to a trace class operator on (Proposition 27.2, [43]) and , since
if , i.e., . It can be easily seen that
and hence . Also if and , then . We can take with , for so that . In addition, if , then .
From now onwards, we use the notation Tr for . We also use , , and , for any in the rest of the paper. Let us recall the commutator estimates of Kato and Ponce [23] used in this paper.
From (2.10), it can be easily seen that for and , the nonlinear term satisfy the estimate:
If u is divergence free, then we have
for all , . Thus it is immediate that
In Remark 2.3, if we take , then we get the following corollary:
For, there exists a constantsuch that, for all,, and, we have
Let us now define the notion of local and maximal strong solutions of the stochastic Euler equations with Lévy noise.
(Local strong solution).
We say that the pair is a local strong (pathwise) solution for the stochastic Euler equations (2.7)–(2.8) if
the symbol τ is a strictly positive stopping time, i.e., ,
for , the symbol u denotes progressively measurable stochastic process such that
, for with , where is the space of all càdlàg paths from to ,
satisfies
satisfies the energy estimate
(Maximal local strong solution).
Let be a predictable process and be a strictly positive stopping time. The pair is said to be a maximal local strong (pathwise) solution for the stochastic Euler equations (2.7)–(2.8), if there exists an increasing sequence with
such that the pair is a local strong solution to (2.7)–(2.8) so that
on the set .
We are now ready to state the main theorem of our paper.
Letbe a given filtered probability space. Letbe the stopping time defined byand let the-measurable initial data, forbe given. Then there exists a local in time strong solutionof the stochastic incompressible Euler equations with Lévy noise ((
2.7
)–(
2.8
)) such that, for any
the energy estimatewhereand,
for a given,where C is a positive constant independent ofuand δ,
,
the-adapted paths ofare càdlàg,
the solutionis pathwise unique,
there exists a unique maximal local strong solution, where.
The truncated stochastic Euler equations
Let us define the Fourier truncation ([17]) as follows:
where , a ball of radius R centered at the origin and is the indicator function. For , we have
where C is a generic constant independent of R.
Let us consider the truncated (in the frequency domain with cut off ) stochastic Euler equations in the whole space as
for . The divergence free condition for can be obtained easily as
and hence . The cut off operator and Helmholtz–Hodge orthogonal projection commutes, since
and hence . On taking the Helmholtz–Hodge orthogonal projection, we get , since
Let us consider the truncated stochastic Euler equations in the Itô stochastic differential form in after taking as
By taking a truncated initial data, we ensure that lie in the space
Note that for . Also, by using (2.12) (see (2.22) also) and Hölder’s inequality, we obtain
for . Thus, we have
and hence is locally Lipschitz in whenever , for . Also, by using (2.16) and the algebra property of -norm, we have
By using (2.28), (2.29) and Theorem 4.9, [29], there exists a pathwise unique strong solution of problem (2.25)–(2.26) in , where depends on R and N (N is defined in (2.67) below). The solution will exist as long as remains finite.
We define a function , for each integer N, as ([25])
Let us consider the truncated stochastic Euler equations with the cut off (denoting ) as
Note that the presence of the cut off function makes the drift term bounded.
Let, for. Then the nonlinear operatorsatisfies
Let , for and for . Then for proving (2.33), we use integration by parts, (2.12) and Hölder’s inequality to the term to get
where in the third step we used the fact that . Thus the estimate (2.34) implies (2.33). □
Energy estimates
Let us first prove the -energy estimate for the stochastic Euler equations (2.25)–(2.26), which is the truncated stochastic Euler equation without the cut off function .
(-energy estimate).
Given the initial datawithbe-measurable, then we havefor anyandfor any.
Let us define the sequence of stopping times to be
Let us apply the Itô’s formula (Theorem 3.7.2, [30], Theorem 4.4, [41], Section 2.3, [32]) to to obtain
for . By using the divergence free condition (see (2.22)) and in , we get
By using the cut off property (2.16), we obtain
On Substituting (2.39) and (2.40) in (2.38), taking expectation, and noting that the stochastic integrals
are local martingales having zero expectation, we get
where we used the fact that . Note that the right hand side of the inequality (2.41) is independent of M. On passing , and then using the dominated convergence theorem in (2.41), we have (2.35).
Let us take supremum from 0 to T in (2.38) and then take expectation to get
where
In the last step of (2.42), we used
since is the Meyer process of and is the quadratic variation process of (Section 2.3, [32]).
Let us apply the Burkholder–Davis–Gundy inequality (Theorem 1.1, [31]), Hölder inequality and Young’s inequality to the term in (2.42) to get
Let us now apply the Burkholder–Davis–Gundy inequality, Hölder inequality and Young’s inequality to the term in (2.42) to obtain
Substituting (2.47) and (2.46) in (2.42), we get
Hence, we have
The right hand side of the inequality (2.48) is independent of M, on passing , and applying the dominated convergence theorem, we get (2.36). □
Proposition 2.9 is also true for the truncated stochastic Euler equation with the cut off function (see (2.31)–(2.32)) as . Note that the -energy estimate for the truncated system (2.25)–(2.26) exists for all .
(-energy estimate).
Let the given initial data, forbe-measurable. Then for any, we haveand for any, we haveand the quantity on the right hand side of the inequality (
2.51
) is independent of R.
An application of the operator on the truncated Eq. (2.31) gives
Let us define the sequence of stopping times to be
Let us apply the Itô’s formula to to obtain
for . Let us consider the term and use the cut off property (2.16) to get
Now let us take the nonlinear term and use the Kato–Ponce commutator estimates (Corollary 2.4) to obtain
where we used in and (2.4). Let us use (2.55) and (2.56) in (2.54) to get
Let us take the expectation in (2.57) to find
where we used (2.45), , and the fact that
are local martingales having zero expectation. Let us apply the Gronwall’s inequality in (2.58) to get
for any . On passing , and using the dominated convergence theorem in (2.59), we get (2.50).
Let us take the supremum from 0 to T in (2.57) and then take the expectation to obtain
Let us take
and
Now by applying the Burkholder–Davis–Gundy inequality, Hölder inequality and Young’s inequality to the term , we get
Once again by applying the Burkholder–Davis–Gundy inequality, Hölder inequality and Young’s inequality to the term , we get
By using (2.61) and (2.62) in (2.60), we obtain
Thus from (2.63), we have
An application of the Gronwall’s inequality yields
Note that the right hand side of the inequality (2.65) is independent of M and R, on passing , we see that . Hence by using the dominated convergence theorem, from (2.65), we finally obtain
where is a constant depending on N and independent of R. □
From Proposition 2.11, it is clear that for the stopping time
the quantity is uniformly bounded and the bound is independent of R. Also the solution of the problem (2.25)–(2.26) can be defined up to a time , where is defined in (2.67) and it can be easily seen that
In the estimate (2.68), we cannot take , as the right hand side of the inequality (2.68) is exponentially growing in N.
Letbe given. Then, we havefor some positive constant C independent of δ.
For the given , there exists a positive integer N such that
Thus, we can choose a δ so that is a local strong solution of (2.25)–(2.26) with
Then it can be easily seen that
where is a positive constant defined by
By using (2.68), we get
where . Then by making use of Markov’s inequality and (2.72), we obtain
where C is a constant independent of u and δ. □
Similar ideas for proving the positivity of stopping time for stochastic quasilinear hyperbolic system can be found in Theorem 1.3 [26].
Existence and uniqueness
We are now ready to prove the existence of local strong solutions of the stochastic Euler equations with Lévy noise (see (2.7)–(2.8)). In order to establish this we first prove that the solutions of smoothed version of the Eqs (2.7)–(2.8) is a Cauchy sequence in the -norm as with probability 1. Let be a local strong solution of the truncated Eqs (2.25)–(2.26), where is the stopping time defined in (2.67).
Letbe-measurable forwith. Then, the family of local strong solutionsof (
2.25
)–(
2.26
) is Cauchy (as) in, i.e.,whereis the stopping time defined in (
2.67
).
Let and be two local strong solutions of (2.25)–(2.26) in and respectively. Let us define and take the difference between the equations for the processes and () to get
Let us apply Itô’s formula to to obtain
for . We write the term from (3.3) as
The third term from the right hand side of the equality (3.4) is zero, by using the divergence free condition (see (2.12) and (2.22)). For and for the stopping time defined in (2.67), let us take the first term from the right hand side of the equality (3.4) and use the Hölder’s inequality, cut off property (see (2.18) and (2.30), with , ) and the algebra property of -norm to obtain
Note that for . By using the Cauchy–Schwarz inequality and Hölder’s inequality, we estimate the second term from the right hand side of the equality (3.4) as
Let us combine (3.5), (3.6) and use it in (3.3) to obtain
Let us take the supremum from 0 to T in (3.7) and then take the expectation to get
where . Let us denote
and
Now we apply the Burkholder–Davis–Gundy inequality, Hölder inequality and Young’s inequality to the term to obtain
By applying the Burkholder–Davis–Gundy inequality, Hölder inequality and Young’s inequality to the term , we get
Substituting (3.9) and (3.10) in (3.8) and then using the inequality , we obtain
Let us take the term from the inequality (3.11) and use (2.18) to find
Let us take final term from the right hand side of the inequality (3.11) and simplify using the cut off property (2.18) to get
for . Hence from (3.11), we have
where . An application of the Gronwall’s inequality in (3.14) yields
Note that the right hand side of the inequality (3.15) is independent of and on passing yields . On passing and applying the dominated convergence theorem, one can easily see that the right hand side of the inequality (3.15) goes to zero and hence the sequence of solutions defined by (by redefining ) is Cauchy (as ) in , where is the stopping time defined in (2.67). □
Letbe-measurable for. Then, the family of local strong solutionsstrongly infor any.
It follows from (3.15) that strongly in . By using the Sobolev’s interpolation inequality (Theorem 9.6, Remark 9.1, [27] with exponents and ) and Hölder’s inequality for , we have
Let us take in (3.16) to obtain
as . Combining Proposition 2.11 and Proposition 3.1 and using the Sobolev’s interpolation yields strongly in for any . □
Since strongly in , we get in , for any and hence implies .
Next let us prove a simple estimate to deal with the nonlinear terms.
Fixand letwith. Then we have
Since, , we have . For , is an algebra and hence
□
Letbe the stopping time defined in (
2.67
), then for, the nonlinear termconverges tostrongly inas.
For , by using (2.16), (2.17), Lemma 3.4 and Hölder’s inequality, for , we obtain
since strongly in and for , . Hence, for the stopping time defined in (2.67), we have strongly in . □
From Proposition 3.5, it can be easily seen that strongly in , since
for the stopping time defined in (2.67).
Letbe the stopping time defined in (
2.67
), then for anyand,in.
By using (2.17) and the Burkholder–Davis–Gundy inequality, for and , we get
as . □
Letbe the stopping time defined in (
2.67
), then for anyand,in.
By using (2.17) and the Burkholder–Davis–Gundy inequality, for and , we obtain
as . □
Since
and
for the stopping time defined in (2.67), we have
for , in .
Letbe the stopping time defined in (
2.67
) and letbe-measurable for. Then there exists a local in time strong solution of problem (
2.7
)–(
2.8
) such that
,
the-adapted paths ofare càdlàg.
From the above construction, we can easily pass the limit as in the equation
for any , so that solves the stochastic Euler equations (2.7)–(2.8) in , for . From Proposition 3.1, Remark 3.6, Proposition 3.7 and Remark 3.10, we conclude that solves the system (2.7)–(2.8) as an equality in , for .
Let us now apply Banach–Alaoglu theorem (Theorem 4.18, [40]) to the Fourier truncated sequence , the solution of the truncated stochastic Euler equations (2.25)–(2.26). From Proposition 2.11, the sequence is uniformly bounded in and the bound is independent of R for the stopping time defined in (2.67). Since is the dual of , the separability of (Remark 10.1.10 and Theorem 10.1.13, [38]) and the uniform bounds for in guarantee the existence of subsequence such that
From the above weak star convergence, the limit u satisfies
and solves (2.7)–(2.8) for . From Proposition 3.1, is almost surely uniformly convergent on finite intervals to , from which it follows that is adapted and càdlàg (Theorem 6.2.3, [1]). □
From the existence of local strong solution to the stochastic Euler equations (2.7)–(2.8), one can easily see that
for the stopping time defined in (2.15) and for any .
Letbe given. Then, for the stopping timedefined in (
2.15
), we havewhere C is a positive constant independent of δ.
Theorem 3.13 can be proved in the same way as that of Theorem 2.13.
Letbe the stopping time defined in (
2.15
) and letbe-measurable for. Let,be two-adapted processes with càdlàg paths that are local strong solutions of (
2.7
)–(
2.8
) having same initial value, such thatfor. Then, a.s., for any, as functions in.
Let be the stopping time defined in (2.15) and let and be two local strong solutions of the system of Eqs (2.7)–(2.8) having common initial data such that . Let us take the difference between the two equations satisfied by and to obtain
Let us apply Itô’s formula to to find
Now we take the nonlinear term, and use the divergence free condition and Hölder’s inequality to get
An application of (3.24) in (3.23) yields
Let us take expectation in (3.25) to get
An application of the Gronwall’s inequality in (3.26) yields
for all . Hence, we get , a.s., for all . In a similar way, one can prove that
for any and then we obtain
Hence, we get , a.s., for any as functions in . □
Letbe the stopping time defined in (
2.15
) and letbe-measurable for. Let,be two-adapted processes with càdlàg paths that are local strong solutions of (
2.7
)–(
2.8
) having same initial data, such thatfor. Then, a.s., for anyas functions in, for any.
Using the Sobolev’s interpolation theorem and Hölder’s inequality for , we find
for any , since , a.s., in . Hence we get , a.s., for any in for any . □
Theorem 3.14 and Theorem 3.15 prove the uniqueness of strong solution of the stochastic Euler equations perturbed by Lévy noise. Now if we take and are two local strong solutions of (2.7)–(2.8), then from (3.24), we get
Thus by Theorem 3.14, we obtain , a.s., for all . A similar calculation to (3.24) also shows that
and hence we find , a.s., for all . Combining the two cases, once can easily see that , a.s., for all and hence , a.s., for all , since is arbitrary. If both are maximal local strong solutions, then it is immediate that , a.s. Since, if is a maximal local strong solution, then , and this must imply that . Otherwise, , by using the equality of , for and the maximality of in and we arrive at a contradiction. A similar argument on imply and hence .
Now, let be the unique local strong solution to the system (2.7)–(2.8) corresponding to the stopping time (2.15). Then, for , by using Theorem 3.14 and Theorem 3.15, we have
since is arbitrary. Thus from the definition of stopping time (2.15), we have , a.s. We can now define , a.s., and
a.s. Hence, is a local strong solution (in fact maximal) to the problem (2.7)–(2.8) and for a given , we have
since (see Theorem 3.13 and Theorem 2.13). A characterization of the maximal solution for stochastic Euler system (2.7)–(2.8) is given in the next theorem (Theorem 3.16).
Assume that Theorem
3.11
and Theorem
3.14
hold. Then there exists a unique pair, which is a maximal strong solution of (
2.7
)–(
2.8
) such thaton the set.
Let us denote by , the set of all stopping times such that if and only if there exists a process such that is a local strong solution of the stochastic Euler equations (2.7)–(2.8). It can be easily seen that
For each , let us take such that be the unique local strong solution of (2.7)–(2.8). Then for each , the process having càdlàg paths such that is a local strong solution of (2.7)–(2.8) with
for some . Now for , let us define a sequence of stopping times such that
Since
it is clear from the definition of that , a.s., for . Thus is a local strong solution of (2.7)–(2.8). But is also a local strong solution of (2.7)–(2.8). Hence from the uniqueness theorem (Theorem 3.14, Theorem 3.15), we get , a.s., for all . This proves that , a.s., for all and hence , a.s., for all . Thus is an increasing sequence in and hence it has a limit in (Proposition 3.9, [6]). Let us denote the limit by . By letting , let be the stochastic process defined by
where . By making use of uniqueness results, we have for any . As , we are thus justified to define a process such that is a local strong solution of (2.7)–(2.8) on the set and hence we have
where is defined in (2.71).
Let us now prove that the maximal solution obtained above is unique. Let us assume that the pair be another maximal solution. Here is an increasing sequence of stopping times converging to and is defined by
By a similar argument above and by the uniqueness theorem (Theorem 3.14 and Theorem 3.15) one can prove that , a.s., for all , for . Let us take so that we get
From (3.41), we can easily conclude that , a.s. If , then either or . In the first case, we have
For , we have
The first identity (3.42) contradicts the fact that does not explode before the time and the second identity contradicts the fact that does not explode before the time . Hence, we must have , a.s., and this proves the uniqueness of the maximal local strong solution of the stochastic Euler equations (2.7)–(2.8). □
Similar ideas of proving maximal local solutions for modeling the flow of liquid crystals can be found in Proposition 3.11, [6] and viscous hydrodynamic systems with multiplicative type of jump noise in Theorem 3.5, [3].
Stochastic Euler equations with multiplicative Lévy noise
The stochastic Euler equations perturbed by multiplicative Lévy noise in (after taking the Helmholtz–Hodge orthogonal projection ) can be written in the Itô stochastic differential form as
where for with and , . We need some additional assumptions on the noise coefficient to prove the existence and uniqueness of local strong solutions to the system (4.1)–(4.2). Let be the space of all Hilbert–Schmidt operators from to ([16]). For an orthonormal basis in , we know that
Let us assume that the noise coefficient and satisfy the following hypothesis of linear growth and Lipschitz condition.
For all, the noise coefficientandsatisfy
(Growth Condition) For alland for all, there exists a positive constant K such that
(Lipschitz Condition) For alland for all, there exists a positive constant L such that
The existence and uniqueness of local strong solutions for the stochastic Euler equations with multiplicative Lévy noise can be proved in the same way as of additive noise case. Proposition 2.11 and Proposition 3.1 can be proved for multiplicative noise case with some changes in the proof due to the presence of in the noise coefficient. Similar estimates in Proposition 2.11 can be obtained with the help of growth condition in Hypothesis 4.1 and the Cauchy sequence result in Proposition 3.1 can be obtained with the help of Lipschitz condition in Hypothesis 4.1. In multiplicative noise case, we have to replace Proposition 3.8 and Proposition 3.9 from Section 3, by the following propositions.
Letbe the stopping time defined in (
2.67
), then for any,and,in.
By using (2.17), Hypothesis 4.1 and the Burkholder–Davis–Gundy inequality, for and , we get
since for any (a proposition similar to Proposition 2.11 in multiplicative noise case) and for any (a proposition similar to Proposition 3.2 in multiplicative noise case). □
Letbe the stopping time defined in (
2.67
), then for any,and,in.
By using (2.17), Hypothesis 4.1 and the Burkholder–Davis–Gundy inequality, for and , we obtain
since for any (a proposition similar to Proposition 2.11 in multiplicative noise case) and for any (a proposition similar to Proposition 3.2 in multiplicative noise case). □
Let us now state the main theorem for incompressible, stochastic Euler equations with multiplicative Lévy noise (see (4.1)–(4.2)).
Letbe a given probability space. Letbe the stopping time defined byand the noise coefficient satisfy Hypothesis
4.1
. Let the-measurable initial data, forbe given. Then, there exists a local in time strong solutionof the problem (
4.1
)–(
4.2
) such that, for any
the energy estimate
for a given,where C is a positive constant independent ofuand δ,
,
the-adapted paths ofare càdlàg,
the solutionis pathwise unique,
there exists a unique maximal local strong solution, where.
Theorem 4.4 can be proved in the same way as of Theorem 3.11.
Footnotes
Acknowledgements
Manil T. Mohan would like to thank National Research Council (NRC) for Research Associateship Award. S.S. Sritharan’s work has been funded by U.S. Army Research Office, Probability and Statistics program. Manil T. Mohan would also like to thank Air Force Institute of Technology (AFIT) for providing stimulating scientific environment and resources.
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