On the Structure of Limiting Flocks in Hydrodynamic Euler Alignment Models
aa r X i v : . [ m a t h . A P ] A ug ON THE STRUCTURE OF LIMITING FLOCKS IN HYDRODYNAMICEULER ALIGNMENT MODELS
TREVOR M. LESLIE AND ROMAN SHVYDKOY
Abstract.
The goal of this note is to study limiting behavior of a self-organized continuous flockevolving according to the 1D hydrodynamic Euler Alignment model. We provide a series of quanti-tative estimates that show how far the density of the limiting flock is from a uniform distribution.The key quantity that controls density distortion is the entropy H = R ρ log ρ d x , and the measureof deviation from uniformity is given by a well-known conserved quantity e = u ′ + L ψ ρ , where u is velocity and L ψ is the communication operator with kernel ψ . The cases of Lipschitz, singulargeometric, and topological kernels are covered in the study. Introduction
In this note we continue the study of the long-time behavior of solutions to the following Euler-alignment model on the torus T = [ − π, π ]:(1) ρ t + ( ρu ) ′ = 0 , (2) u t + uu ′ = Z T ψ ( x, y )( u ( y ) − u ( x )) ρ ( y ) d y = L ψ ( ρu ) − u L ψ ( ρ ) , where L ψ f := Z T ψ ( x, y )( f ( y ) − f ( x )) d y, and ψ : T → [0 , ∞ ) is a communication kernel. (Here and below, we use primes to denotespatial derivatives: f ′ = ∂ x f .) This model represents a one-dimensional hydrodynamic analogue ofthe Cucker-Smale agent based dynamical system [3, 4], and found application in a wide variety ofsubjects, see [8, 14, 1] for recent surveys. The system (1) – (2) is designed to describe the mechanismof alignment of congregations of agents governed by laws of self-organization with communicationencoded into the kernel ψ . The long time behavior is thus characterized by convergence to a flockingstate, by which we mean alignment to a constant velocity u → ¯ u , and stabilization of density to atraveling wave(3) ρ ( x, t ) → ρ ∞ ( x − t ¯ u ) . Such a result was proved under the strong global communication condition inf ψ > R ∞ ψ ( r ) d r = ∞ in a variety of settings,[2, 13, 6]. The case of local kernels, by which we mean purely local protocols, supp ψ ⊂ {| x − y | R } ,remains largely open with the exception of a new class of topological kernels introduced in [9], andthe case of strong communication relative to other initial parameters of the data, [7]. Date : August 15, 2019.1991
Mathematics Subject Classification.
Key words and phrases. flocking, alignment, collective behavior, emergent dynamics, fractional dissipation, Cucker-Smale.
Acknowledgment.
RS was supported in part by NSF grants DMS-1515705, DMS-1813351, and the SimonsFoundation. TL was supported in part by NSF grants DMS-1147523 (PI: Andreas Seeger) and DMS 1515705 (PI:Roman Shvydkoy).
The multitude of flocking states (3) demonstrates that the Euler alignment system (1) – (2)supports a variety of self-organization outcomes. However it is hard to predict what that outcome ρ ∞ will be from initial conditions. Note that ¯ u , on the other hand, is uniquely determined bythe ratio of conserved momentum over mass. In this article we propose to study this questionwith a less ambitious goal: determine how far the limiting flock ρ ∞ deviates from the uniformdistribution ¯ ρ = π M , where M is the total mass. In fact, we consider a more general case whenthe convergence (3) is unknown. As a measure of “disorder” of the flock we consider the long timelimit(4) lim sup t →∞ k ρ ( · , t ) − ¯ ρ k L ( T ) . Let us recall that in 1D the Euler alignment system possesses an extra conserved quantity (see[13, 2, 9]) e t + ( ue ) ′ = 0 , e = u ′ + L ψ ρ, provided ψ is either of convolution type, ψ ( x, y ) = ψ ( x − y ), or topological type as defined below.The physical nature of this quantity has remained elusive, but we will find that it is directlyimplicated in quantifying disorder of the flock similar to the topological entropy. More precisely,we consider the e -quantity per mass of the flock: q = eρ . Note that q is transported:(5) q t + uq ′ = 0 . This allows to trace information at any time t back to the initial datum, in particular, k q ( · , t ) k ∞ = k q k ∞ . The thrust of our main results is to show that the latter is the parameter that controlsdeviation from the uniform flock expressed by the limit (4).Let us set some assumptions. We distinguish two classes of kernels: • Lipschitz convolution type kernels ψ ∈ Lip( T ) with local communication(6) ψ ( x − y ) > λχ R ( | x − y | ) , R , λ > • Symmetric topological kernels, as introduced in [9] : here τ >
0, 0 < α < ψ ( x, y, t ) = h ( x − y ) | x − y | α − τ d( x, y, t ) τ , h ( x − y ) > λχ R ( | x − y | ) , where d( x, y, t ) = (cid:12)(cid:12)(cid:12)(cid:12)Z yx ρ ( z, t ) d z (cid:12)(cid:12)(cid:12)(cid:12) . The well-posedness theory for the smooth case was developed in [2, 13], and for the singular case in[5, 10, 11, 12] (geometric kernels, τ = 0) and [9] (topological kernels, τ > u ′ + ψ ∗ ρ > τ α and for small data if τ > α .We now state our main results. Theorem 1.1.
Let ( ρ, u ) be a smooth solution to the system (1) – (2) , with kernel given by either (6) or (7) . If e = 0 , then (8) k ρ ( t ) − ¯ ρ k L c ( k ρ k L ) e − λc ( R ,M ,α,τ, k ρ k L ∞ ) t , where c depends only on R and M in the Lipschitz case, and c need not depend on k ρ k L ∞ if τ α + 1 in the topological case. We cite here the first draft of the manuscript [9]; in later versions the authors specialize to the case where τ = n ,where n is the dimension of the space. N THE STRUCTURE OF LIMITING FLOCKS IN HYDRODYNAMIC EULER ALIGNMENT MODELS 3
We note that this result complements the one obtained in [9] for topological case. Namely, if e = 0, then in the L ∞ metric one has a slower algebraic relaxation towards the uniform state: k ρ ( t ) − ¯ ρ k L ∞ . √ t , as t → ∞ . Theorem 1.2.
Let ( ρ, u ) be a smooth solution to the system (1) – (2) , with a Lipschitz kernel ψ satisfying (6) . Provided k q k L ∞ < k ψ k L , one has (9) lim sup t →∞ k ρ ( · , t ) − ¯ ρ k L M k q k L ∞ k ψ k L ∞ λc ( R )( k ψ k L − k q k L ∞ ) , where c ( R ) is a constant depending only on R . Let us note that the dependence on k q k L ∞ is linear for small values. At the same time, the boundis inversely proportional to the strength λ , which shows the stabilizing effect of communication onthe structure of the flock. Theorem 1.3.
Let ( ρ, u ) be a smooth solution to (1) – (2) , with topological kernel ψ . One has thefollowing bounds for any initial data lim sup t →∞ k ρ ( · , t ) − ¯ ρ k L ( cλ − M τ k q k L ∞ [( α − τ ) λ − M τ k q k L ∞ + R τ − α ] α − τ , τ < αcλ − M τ k q k L ∞ exp (cid:0) λ − M τ k q k L ∞ (cid:1) , τ = α, where c = c ( R , α, τ ) . And one has the following two bounds lim sup t →∞ k ρ ( · , t ) − ¯ ρ k L ( cλ − M τ k q k L ∞ [( α − τ ) λ − M τ k q k L ∞ + R τ − α ] α − τ α < τ < αcλ − M τ k q k L ∞ [( α − τ ) λ − M τ k q k L ∞ + R τ − α ] α − τ τ > α, under the smallness requirement k q k L ∞ < λM − τ R τ − α τ − α . Let us note that for small q all the bounds are essentially linear in k q k L ∞ . Other factorsminimize distortions of the flock as well, such as strength of communication λ >>
1, or small mass.
Remark . We actually only need lower bounds on q in the smallness conditions of the Theoremsabove. For example, in (9), the quantity k ψ k L − k q k L ∞ can be replaced with k ψ k L + inf q ; onecan see this by following the estimates on the density amplitude in Section 2.5. This results in theslightly relaxed smallness requirement inf q > −k ψ k L . A similar remark applies to the statementfor topological kernels. However, since our primary interest is in the case where k q k L ∞ is small,we state our results entirely in terms of k q k L ∞ , for simplicity.Note also that in the case of Lipschitz kernels, inf q > −k ψ k L (with non-strict inequality)is equivalent to the critical threshold condition (c.f. [2], [10]) that dictates whether the solutionremains globally smooth or blows up in finite time. So the precise version of the smallness conditionwe need in order to control the density is almost guaranteed already by the fact that we are workingwith globally smooth solutions. The situation is similar for topological kernels in the case τ > α :the smallness condition is precisely the one from the existence theory in [9]. Unlike the case ofLipschitz kernels, however, it is not known how sharp this condition is for the existence theory fortopological kernels. TREVOR M. LESLIE AND ROMAN SHVYDKOY Proofs of the results
Let us recall some preliminary facts before we proceed to the main proofs.2.1.
The Csisz´ar-Kullback inequality.
The main tool in establishing results of this paper isthe use of relative entropy defined by(10) H = Z T ρ log ρ ¯ ρ d x = Z T ρ log ρ d x − M log ¯ ρ, where ¯ ρ = π M and M = R T ρ ( x ) d x is the total mass. The classical Csisz´ar-Kullback inequalitystates(11) k ρ − ¯ ρ k L π ¯ ρ H . Furthermore, by the elementary inequality log x x −
1, we also have(12) H Z T ρ (cid:18) ρ ¯ ρ − (cid:19) d x = ¯ ρ − k ρ − ¯ ρ k L . Thus we obtain the two sided bounds(13) 14 π k ρ − ¯ ρ k L ¯ ρ H k ρ − ¯ ρ k L . Evolution of the entropy.
At the heart of the argument is the equation on the entropy (10)which one obtains testing the continuity equation (1) with log ρ + 1:( ρ log ρ ) t = ρ t (log ρ + 1) = − ( ρu ) ′ (log ρ + 1)= − ρ ′ (log ρ + 1) u − ρu ′ (log ρ + 1)= − ( ρ log ρ ) ′ u − ( ρ log ρ ) u ′ − ρu ′ = − [ u ( ρ log ρ )] ′ − ρu ′ = − [ u ( ρ log ρ )] ′ − ρ q + ρ L ψ ρ. Therefore,(14) d H d t = dd t Z T ρ log ρ d x = − Z T ρ q d x − Z T ψ ( x, y )( ρ ( x ) − ρ ( y )) ρ ( x ) d x d y. Noting that R T ρq d x = R T e d x = 0, we can subtract ¯ ρ from one density in the first integral on theleft hand side. After additionally symmetrizing the last integral we obtain(15) d H d t = − Z T ( ρ − ¯ ρ ) ρq d x − Z T ψ ( x, y ) | ρ ( x ) − ρ ( y ) | d x d y. Bounds on the dissipation.
If our kernels ψ were global, it would be easy to get a positivelower bound on the dissipation term:(16) Z T ψ ( x, y ) | ρ ( x ) − ρ ( y ) | d x d y > (inf ψ ) Z T | ρ ( x ) − ρ ( y ) | d x d y = 2(inf ψ ) k ρ − ¯ ρ k L . Since in all our cases we have a non-trivial lower bound on the kernel only near the diagonal { ( x, y ) ∈ T : | x − y | < R } , we need a substitute for (16) stated in the following Lemma. Lemma 2.1.
The following inequality holds: (17) 12 Z T Z | z |
Proof.
Let χ be a nonnegative cutoff function on T with support in B R (0), which is constant on B R / and has integral 1. Then b χ (0) = 1 and | b χ ( k ) | < k ∈ Z \{ } . On the other hand, wehave by the Riemann-Lebesgue Lemma that b χ ( k ) → k → ∞ ; therefore we have in fact that | b χ ( k ) | − ε for some ε > R ( k = 0). Define ¯ ρ R ( x ) = χ ∗ ρ ( x ), so that( ρ − ¯ ρ R ) b ( k ) = (1 − b χ ( k )) b ρ ( k ) . Consequently | ( ρ − ¯ ρ R ) b ( k ) | > ε | b ρ ( k ) | , k ∈ Z , k = 0 , and b ρ (0) = b ¯ ρ R (0). Thus k ρ − ¯ ρ k L = X k ∈ Z \{ } | b ρ ( k ) | ε − X k ∈ Z | ( ρ − ¯ ρ R ) b ( k ) | = ε − k ρ − ¯ ρ R k L . By the fact that R T χ = 1 and Minkowski, we have k ρ − ¯ ρ R k L = (cid:13)(cid:13)(cid:13)(cid:13)Z T χ ( y )( ρ ( · ) − ρ ( · − y )) d y (cid:13)(cid:13)(cid:13)(cid:13) L Z | y |
It turns out that the choice of η ∈ [0 , min { α, τ } ] in (22) has very little effect on the result ofTheorem 1.3. Eventually, we will set η = τ or η = 1 + α , simply to clean up some of the exponentsthat appear in our bounds; however, we will carry along the η -dependence to show that we don’tlose any information by making these choices. The only place where the lower bound (21) givesan advantage over the more general (22) is in the proof of Theorem 1.1. In the case e = 0, thebound (22) is uniform in time, with k ρ k L ∞ replacing k ρ ( t ) k L ∞ . As a result, the constant c fromTheorem 1.1 carries a dependence on k ρ k τ − ηL ∞ . If τ α , then we can choose η = τ (i.e., use(21) instead of (22)) and eliminate this dependency.Finally, we note that for the purposes of bounding the density amplitude, using η = τ in (20)will be crucial, in contrast to the computations on which (22) depends. Our density bounds willrely on the full kernel ψ ( x, y, t ) rather than just its lower bound ψ ( t ) near the diagonal.2.4. The entropy equation revisited.
We now return to the evolution equation (15) and makeestimates; in doing so we will prove Theorem 1.1, and we will reduce the proofs of Theorems 1.2and 1.3 to proving the relevant bounds on the density amplitude.In what follows, the constant c = c ( R ) depends only on R but may change from line to line.We have ˙ H ( t ) k ρ ( t ) k L ∞ k q k L ∞ k ρ ( · , t ) − ¯ ρ k L − c ( R ) ψ ( t ) k ρ ( · , t ) − ¯ ρ k L k ρ ( t ) k L ∞ k q k L ∞ p π ¯ ρ H ( t ) − c ( R ) ψ ( t )¯ ρ H ( t ) . Setting Y = √H , we find ˙ Y ( t ) k ρ ( t ) k L ∞ k q k L ∞ √ π ¯ ρ − c ( R ) ψ ( t )¯ ρY ( t ) . Using Gr¨onwall’s lemma we obtain(23) Y ( t ) Y exp (cid:18) − c ( R )¯ ρ Z t ψ ( s ) d s (cid:19) + √ π ¯ ρ k q k L ∞ Z t k ρ ( s ) k L ∞ exp (cid:18) − c ( R )¯ ρ Z ts ψ ( τ ) d τ (cid:19) d s. From here, it is easy to prove Theorem 1.1. Indeed, if e ≡
0, then the second term in (23) dropsout completely. Furthermore, ρ satisfies a maximum principle in this case: k ρ ( t ) k L ∞ k ρ k L ∞ , so ψ is uniformly bounded below by a constant in all cases: ψ ( s ) > ψ . k ρ ( · , t ) − ¯ ρ k L p π ¯ ρY ( t ) p π ¯ ρ H exp( − c ( R )¯ ρtψ ) √ π k ρ − ¯ ρ k L exp( − λc t ) c k ρ k L exp( − λc t ) , where c = c ( R )¯ ρ ( ψ/λ ). Thus, in the Lipschitz case, c depends only on R and M . In thetopological case, c depends only on R , M , α , and τ when τ α ; if τ > α , then c depends additionally on k ρ k L ∞ . This completes the proof of Theorem 1.1.If ψ is a Lipschitz kernel or a topological kernel with τ α , then we know that ψ ( t ) isuniformly bounded below, hence the first term on the right side of (23) vanishes as t → ∞ . In fact,the same is true for topological kernels with τ > α ; this will be apparent once we prove upperbounds on the density amplitude. We write(24) lim sup t →∞ k ρ ( · , t ) − ¯ ρ k L M k q k L ∞ lim sup t →∞ Z t k ρ ( s ) k L ∞ exp (cid:18) − c ( R )¯ ρ Z ts ψ ( τ ) d τ (cid:19) d s. If ψ is a Lipschitz kernel, then this estimate becomes simply(25) lim sup t →∞ k ρ ( · , t ) − ¯ ρ k L k q k L ∞ λc ( R ) lim sup t →∞ k ρ ( t ) k L ∞ . N THE STRUCTURE OF LIMITING FLOCKS IN HYDRODYNAMIC EULER ALIGNMENT MODELS 7
Thus the proof of Theorem 1.2 is reduced to estimating the density amplitude. The situation issimilar for topological kernels. Upon substituting (22) into (24), the usefulness of the followingLemma becomes apparent:
Lemma 2.2.
Suppose f is a positive bounded function on R + with lim sup t →∞ f ( t ) = L . Let c > , δ > . Then lim sup t →∞ Z t f ( s ) exp (cid:26) − Z ts cf δ ( l ) d l (cid:27) d s L δ c . Proof.
Fix ε >
0, and T so that f ( t ) < L + ε for t > T . Since f is bounded, it is clear that Z T f ( s ) exp (cid:26) − Z ts cf δ ( l ) d l (cid:27) d s → , as t → ∞ . Let us estimate the rest Z tT f ( s ) exp (cid:26) − Z ts cf δ ( l ) d l (cid:27) d s Z tT ( L + ε ) exp (cid:26) − ( t − s ) c ( L + ε ) δ (cid:27) d s ( L + ε ) δ c . This finishes the proof. (cid:3)
Applying the Lemma to (24) and (22), we conclude that for any η ∈ [0 , min { τ, α } ], we have(26) lim sup t →∞ k ρ ( · , t ) − ¯ ρ k L M η R α − η k q k L ∞ λc ( R ) lim sup t →∞ k ρ ( t ) k τ − ηL ∞ . Thus in all cases, all that remains is a large-time bound on the density amplitude.2.5.
Bounds on the Density Amplitude.
Throughout our discussion of bounds on the densityamplitude, we will make use of the following differential inequality: If f ˙ X ( t ) AX ( t )[ B − X ( t )],where A and B are positive constants and X ( t ) is a positive function, then(27) X ( t ) BX (0) X (0) + ( B − X (0)) exp( − ABt ) . In particular, lim sup t →∞ X ( t ) B .2.5.1. Case of Lipschitz kernels.
Let ρ + ( t ) denote the maximum value of ρ at time t , and let x + denote the x -value where the maximum is achieved. Then if k q k L ∞ < k ψ k L , one can get an upperbound on k ρ ( t ) k L ∞ by integrating the differential inequality derived below.dd t ρ + ( t ) = − ρ + ( t ) u ′ ( x + , t ) = − ρ + ( t ) q ( x + , t ) + ρ + ( t ) Z T ψ ( x + − y )( ρ ( y, t ) − ρ + ( t )) d y ( k q k L ∞ − k ψ k L ) ρ + ( t ) + k ψ k L ∞ M ρ + ( t )= ( k ψ k L − k q k L ∞ ) ρ + ( t ) (cid:20) k ψ k L ∞ M k ψ k L − k q k L ∞ − ρ + ( t ) (cid:21) . In view of (27) we obtain lim sup t →∞ k ρ ( t ) k L ∞ k ψ k L ∞ M k ψ k L − k q k L ∞ . Plugging into (25) we concludelim sup t →∞ k ρ ( · , t ) − ¯ ρ k L M k q k L ∞ k ψ k L ∞ λc ( R )( k ψ k L − k q k L ∞ ) . TREVOR M. LESLIE AND ROMAN SHVYDKOY
Case of topological kernels with τ α . The case of topological kernels follows thesame strategy as for Lipschitz kernels, with additional technicalities. Note that in order for thedissipation to compete with the quadratic term qρ , we must choose η = τ in (20); otherwise theassociated power of ρ + will be less than 2.For any r ∈ (0 , R ), we havedd t ρ + ( t ) = − q ( x + , t ) ρ + ( t ) + ρ + ( t ) Z ψ ( x + , x + + z )( ρ ( x + + z, t ) − ρ + ( t )) d z k q k L ∞ ρ + ( t ) + λM − τ ρ + ( t ) Z r< | z |
If this is the case, then it follows that there exists r > I ( r ) = λ − M τ k q k L ∞ and I ( r ) > λ − M τ k q k L ∞ for r ∈ (0 , r ). The value of this r is given by r τ − α = R τ − α − ( τ − α ) λ − M τ k q k L ∞ . If r τ − α = (1 + α − τ )[ R τ − α − ( τ − α ) λ − M τ k q k L ∞ ] = (1 + α − τ ) r τ − α , then clearly r ∈ (0 , r ). Furthermore, this formula agrees with that given in (29) (for τ < α ), so bythe same manipulations as before, we havelim sup t →∞ k ρ ( t ) k L ∞ r − − α + τ M I ( r ) − λ − M τ k q k L ∞ = (1 + α − τ ) M r − = (1 + α − τ ) − τ − α M [ R τ − α − ( τ − α ) λ − M τ k q k L ∞ ] α − τ Plugging this into (26) we obtainlim sup t →∞ k ρ ( · , t ) − ¯ ρ k L M τ k q k L ∞ λR η c ( R , α, τ )[( α − τ ) λ − M τ k q k L ∞ + R τ − α ] τ − ητ − α . Once again, we choose η = τ to obtain the bound in Theorem 1.3.2.5.4. Case of topological kernels with τ > α + 1 . When τ > α + 1, we still require(31) λ − M τ k q k L ∞ < I (0) = R τ − α / ( τ − α )in order to get an upper bound on the density. However, our initial estimate on the time derivativeof ρ + ( t ) needs minor adjustments, since the power 1 + α − τ associated to the geometric part ofthe kernel is no longer positive:dd t ρ + ( t ) k q k L ∞ ρ + ( t ) + λM − τ ρ + ( t ) Z | z | Active particles. Vol. 1. Advances in theory, models, and applications , Model.Simul. Sci. Eng. Technol., pages 259–298. Birkh¨auser/Springer, Cham, 2017.[2] Jos´e A. Carrillo, Young-Pil Choi, Eitan Tadmor, and Changhui Tan. Critical thresholds in 1D Euler equationswith non-local forces. Math. Models Methods Appl. Sci. , 26(1):185–206, 2016.[3] Felipe Cucker and Steve Smale. Emergent behavior in flocks. IEEE Trans. Automat. Control , 52(5):852–862,2007.[4] Felipe Cucker and Steve Smale. On the mathematics of emergence. Jpn. J. Math. , 2(1):197–227, 2007.[5] Tam Do, Alexander Kiselev, Lenya Ryzhik, and Changhui Tan. Global Regularity for the Fractional EulerAlignment System. Arch. Ration. Mech. Anal. , 228(1):1–37, 2018. [6] Seung-Yeal Ha and Eitan Tadmor. From particle to kinetic and hydrodynamic descriptions of flocking. Kinet.Relat. Models , 1(3):415–435, 2008.[7] Javier Morales, Jan Peszek, and Eitan Tadmor. Flocking With Short-Range Interactions. J. Stat. Phys. ,176(2):382–397, 2019.[8] Sebastien Motsch and Eitan Tadmor. Heterophilious dynamics enhances consensus. SIAM Rev. , 56(4):577–621,2014.[9] R. Shvydkoy and E. Tadmor. Topological models for emergent dynamics with short-range interactions. ArXive-prints , June 2018.[10] Roman Shvydkoy and Eitan Tadmor. Eulerian dynamics with a commutator forcing. Transactions of Mathematicsand Its Applications , 1(1), 2017.[11] Roman Shvydkoy and Eitan Tadmor. Eulerian dynamics with a commutator forcing II: Flocking. Discrete Contin.Dyn. Syst. , 37(11):5503–5520, 2017.[12] Roman Shvydkoy and Eitan Tadmor. Eulerian dynamics with a commutator forcing III: Fractional diffusion oforder 0 < α < Physica D , 2017.[13] Eitan Tadmor and Changhui Tan. Critical thresholds in flocking hydrodynamics with non-local alignment. Philos.Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , 372(2028):20130401, 22, 2014.[14] Tams Vicsek and Anna Zafeiris. Collective motion. Physics Reports , 517(3):71 – 140, 2012. Department of Mathematics, University of Wisconsin, Madison E-mail address : [email protected] Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago E-mail address ::