LONG-TIME BEHAVIOUR OF INTERACTION MODELS ON RIEMANNIAN MANIFOLDS WITH BOUNDED CURVATURE RAZVAN C. FETECAU AND HANSOL PARK

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LONG-TIME BEHAVIOUR OF INTERACTION MODELS ON
RIEMANNIAN MANIFOLDS WITH BOUNDED CURVATURE
RAZVAN C. FETECAU AND HANSOL PARK
Abstract. We investigate the long-time behaviour of solutions to a nonlocal partial dif-
ferential equation on smooth Riemannian manifolds of bounded sectional curvature. The
equation models self-collective behaviour with intrinsic interactions that are modelled by
an interaction potential. We consider attractive interaction potentials and establish suffi-
cient conditions for a consensus state to form asymptotically. In addition, we quantify the
approach to consensus, by deriving a convergence rate for the diameter of the solution’s
support. The analytical results are supported by numerical simulations for the equation
set up on the rotation group.
1. Introduction
In this paper we investigate the long-time behaviour of measure-valued solutions to the
following integro-differential equation on a Riemannian manifold M:
tρ+M·(ρv)=0,(1.1a)
v=−∇MKρ,(1.1b)
where K:M×MRis an interaction potential, and M·and Mrepresent the manifold
divergence and gradient, respectively. In (1.1b) the symbol denotes a measure convolution:
for a time-dependent measure ρton Mand xMwe set
Kρt(x) = ZM
K(x, y) dρt(y).(1.2)
Equation (1.1a) is in the form of a continuity equation that governs the transport of the
measure ρalong the flow on Mgenerated by the velocity field vgiven by (1.1b). Note
that (1.1a) is an active transport equation, as the velocity field has a nonlocal functional
dependence on ρitself. This geometric interpretation plays a major role in the paper, as
measure-valued solutions to (1.1) are defined via optimal mass transport [10, 25]. Also,
since (1.1) conserves the total mass, we restrict our solutions to be probability measures on
Mat all times, i.e., RMdρt= 1 for all t0. We note that system (1.1) has a discrete/ODE
analogue that has an interest in its own [11], and the general framework of measure-valued
solutions includes the discrete formulation as well.
In the literature, equation (1.1) is interpreted as an aggregation or interaction model, with
ρrepresenting the density of a certain population. Indeed, the velocity vat xcomputed by
(1.1b) accounts for all contributions from interactions with point masses yin the support of
Date: October 20, 2022.
2020 Mathematics Subject Classification. 35A30, 35B38, 35B40, 58J90.
Key words and phrases. measure solutions, intrinsic interactions, asymptotic consensus, swarming on
manifolds.
1
arXiv:2210.10153v1 [math.AP] 18 Oct 2022
2 FETECAU AND PARK
ρ, through the convolution. As a result of this interaction, point xmoves either toward or
away from y, depending whether the interaction with yis attractive or repulsive. The nature
of the interaction between xand yis set by the vector −∇MK(x, y) (the gradient is taken
with respect to x), which also provides the direction and magnitude of their interaction
[26, 24, 25]. With such interpretation, model (1.1) has numerous applications in swarming
and self-organized behaviour in biology [13], material science [14], robotics [27, 32], and
social sciences [38]. Depending on the application, equation (1.1) can model interactions
between biological organisms (e.g., insects, birds or cells), robots or even opinions.
While there is extensive recent analytical and numerical work on solutions to model (1.1),
this research has focused almost exclusively on the model set up on the Euclidean space Rn.
For analysis of (1.1) in Euclidean setups we refer to [8, 6, 12, 7] for the well-posedness of the
initial-value problem, to [35, 20, 5, 23, 22] for the long-time behaviour of its solutions, and
to [3, 18, 41] for studies on minimizers for the associated interaction energy. Numerically,
it has been shown that model (1.1) can capture a wide variety of self-collective or swarm
behaviours, such as aggregations on disks, annuli, rings and soccer balls [34, 44, 43].
The literature on solutions to model (1.1) set up on general Riemannian manifolds is
very limited, with only a few works on this subject. In this respect we distinguish two
approaches: extrinsic and intrinsic. In the extrinsic approach the manifold Mis assumed
to have a natural embedding in a larger Euclidean space, and interactions between points on
Mdepend on the Euclidean distance in the ambient space between the points [46, 15, 39].
The second approach considers intrinsic interactions, which only depend on the intrinsic
geometry of the manifold [25, 26]. In particular, the interaction potential can be assumed in
this case to depend on the geodesic distance on the manifold between points [25, 26]. The
goal of the present paper is to consider such interaction potentials and take a fully intrinsic
approach to study the long time behaviour of solutions to (1.1) on general Riemannian
manifolds with bounded sectional curvature.
Well-posedness of measure-valued solutions to equation (1.1) set up on general Riemann-
ian manifolds, with intrinsic interactions, has been established recently in [25]. Previous
works considered the equation on particular manifolds, such as sphere and cylinder [24]
and the special orthogonal group SO(3) [21]. The long-time behaviour of solutions to (1.1)
on manifolds has also been considered recently. Rich pattern formation behaviours have
been shown for the model with intrinsic interactions on the sphere and the hyperbolic plane
[26, 24], and on the special orthogonal group SO(3) [21]. For the extrinsic approach, emer-
gent behaviour has been studied on various manifolds such as sphere [17], unitary matrices
[36, 31, 30], hyperbolic space [28], and Stiefel manifolds [29].
In the present research we consider purely attractive potentials and investigate the long
time behaviour of the solutions to (1.1). For strongly attractive potentials, the focus is on the
formation of consensus solutions, where the equilibria consist of an aggregation at a single
point. In the engineering literature, achieving such a state is also referred to as synchro-
nization or rendezvous. Bringing a group of agents/robots to a rendezvous configuration is
an important problem in robotic control [40, 42, 37]. We also note here that applications of
(1.1) in engineering (robotics) often require a manifold setup, as agents/robots are typically
restricted by environment or mobility constraints to remain on a certain manifold. In such
applications, for an efficient swarming or flocking, agents must approach each other along
geodesics, further motivating the intrinsic approach taken in this paper. The emergence
of self-synchronization has also numerous occurrences in biological, physical and chemical
INTERACTION MODELS ON MANIFOLDS WITH BOUNDED CURVATURE 3
systems (e.g., flashing of fireflies, neuronal synchronization in the brain, quantum synchro-
nization) – see [36, 28] and references therein. For applications of asymptotic consensus to
opinion formation, we refer to [38].
Emergence of asymptotic consensus in intrinsic interaction models on Riemannian mani-
folds has been studied recently for certain specific manifolds such as sphere [24] and rotation
group [21], as well as for general manifolds of constant curvature [21]. In the current paper
we take a very general approach and investigate the formation of consensus on arbitrary
manifolds of bounded curvature. We establish sufficient conditions on the interaction po-
tential and on the support of the initial density for a consensus state to form asymptotically.
Compared to the most general results available to date [21], the present research improves in
several key aspects (see Remark 4.2): i) considers general manifolds of bounded curvature,
ii) relaxes the assumptions on the interaction potential, and iii) provides a quantitative rate
of convergence to consensus. In particular, by relaxing the assumption on K, our study
includes now the important class of power-law potentials. We also provide numerical ex-
periments for M=SO(3) and show that the analytical rate of convergence to consensus is
sharp.
The paper is structured as follows. In Section 2, we present some preliminaries on the
interaction equation (1.1) and on some results from Riemannian geometry; in particular,
we set the notion of the solution and the assumptions on the interaction potential. In
Section 3, we prove the formation of asymptotic consensus for strongly attractive potentials
(Theorem 3.1). With an additional assumption on the interaction potential, in Section 4 we
quantify the approach to consensus by establishing a rate of convergence for the diameter
of the support of the solution (Theorem 4.1). In Section 5 we consider weakly attractive
potentials and investigate the asymptotic behaviour (Theorem 5.1). In Section 6, we present
some numerical results for M=SO(3). Finally, the Appendix includes some additional
comments on well-posedness and the proofs of several lemmas.
2. Preliminaries
In this section, we introduce the notion of solutions to model (1.1) and discuss the well-
posedness of solutions, as established in [25]. We then present the gradient flow formulation
of model (1.1), and some concepts and results from Riemannian geometry that we will use
in the paper.
The following assumptions on the manifold Mand interaction potential Kwill be made
throughout the entire paper.
(M)Mis a complete, smooth Riemannian manifold of finite dimension n, with positive
injectivity radius inj(M)>0. We denote its intrinsic distance by dand sectional curvature
by K.
(K) The interaction potential K:M×MRhas the form
K(x, y) = g(d(x, y)2),for all x, y M,
where g: [0,)Ris differentiable, g0is locally Lipschitz continuous, and
(2.3) g0(r2)0,for all 0 < r < inj(M).
In particular, (2.3) indicates that the potential Kis attractive – see explanation below.
Anywhere in the paper, ·denotes the inner product of two tangent vectors (in the same
tangent space) and k · k represents the norm of a tangent vector. The tangent bundle of
4 FETECAU AND PARK
Mis denoted by T M. We will also omit the subscript Mon the manifold gradient and
divergence.
The expression (1.1b) for v(see also (1.2)) involves the gradient of K, which in turn is a
function of the squared distance function d2. For all x, y Mwith d(x, y)<inj(M), the
gradient with respect to xof d2is given by
(2.4) xd(x, y)2=2 logxy,
where logxydenotes the Riemannian logarithm map (i.e., the inverse of the Riemannian
exponential map) [19]. Hence, by chain rule we have
(2.5) xK(x, y) = 2g0(d(x, y)2) logxy.
The velocity at x, as computed by (1.1b), considers all interactions with point masses
ysupp(ρ). By (2.5), when a point mass at xinteracts with a point mass at y(we assume
here d(x, y)<inj(M)), the mass at xis driven by a force of magnitude proportional to
|g0(d(x, y))2|d(x, y), to move either towards y(provided g0(d(x, y)2)>0) or away from y
(provided g0(d(x, y)2)<0). If g0(x, y) = 0, the two point masses do not interact at all. For
a potential that satisfies (K), any two point masses within inj(M) distance from each other
either feel an attractive interaction, or do not interact at all.
2.1. Notion of the solution and well-posedness. Denote by UMa generic open
subset of M, and by P(U) the set of Borel probability measures on the metric space (U, d).
Also denote by C([0, T ); P(U)) the set of continuous curves from [0, T ) into P(U) endowed
with the narrow topology (i.e., the topology dual to the space of continuous bounded func-
tions on U; see [2]).
For Ψ : Σ U, with Σ Ua measurable set, we denote by Ψ#ρthe push-forward in the
mass transport sense of ρthrough Ψ. Equivalently, Ψ#ρis the probability measure such
that for every measurable function ϕ:U[−∞,] with ϕΨ integrable with respect to
ρ, it holds that
(2.6) ZU
ϕ(x) d(Ψ#ρ)(x) = ZΣ
ϕ(Ψ(x)) dρ(x).
We define solutions to (1.1) in a geometric way, as the push-forward of an initial density ρ0
through the flow map on Mgenerated by the velocity field vgiven by (1.1b) [2, Chapter 8.1].
To set up terminology, consider a time-dependent vector field X:U×[0, T )T M and a
measurable subset Σ U. The flow map generated by (X, Σ) is a function ΨX: Σ×[0, τ)
U, for some τT, that for all xΣ and t[0, τ) satisfies
(2.7)
d
dtΨt
X(x) = Xtt
X(x)),
Ψ0
X(x) = x,
where we used the notation Xtto denote X(·, t) and Ψt
Xfor ΨX(·, t). A flow map is said to
be maximal if its time domain cannot be extended while (2.7) holds; it is said to be global
if τ=T=and local otherwise.
For T > 0 and a curve (ρt)t[0,T )⊂ P(U), the vector field in model (1.1) is vgiven by
(1.1b). To indicate its dependence on ρ, let us rewrite (1.1b) as
(2.8) v[ρ](x, t) = −∇Kρt(x),for (x, t)U×[0, T ),
where for convenience we used ρtin place of ρ(t), as we shall often do in the sequel.
We adopt the following definition of solutions to equation (1.1).
INTERACTION MODELS ON MANIFOLDS WITH BOUNDED CURVATURE 5
Definition 2.1 (Notion of weak solution).Given UMopen, we say that (ρt)t[0,T )
P(U)is a weak solution to (1.1) if (v[ρ],supp(ρ0)) generates a unique flow map Ψv[ρ]defined
on supp(ρ0)×[0, T ), and ρtsatisfies the implicit transport equation
ρt= Ψt
v[ρ]#ρ0,for all t[0, T ).
Note that by Lemma 8.1.6 of [2], a weak solution to (1.1) defined as above is also a weak
solution in the sense of distributions. The local well-posedness of solutions to model (1.1)
(in the sense of Definition 2.1) was established in [25, Theorem 4.6]. For purely attractive
potentials, the local well-posedness can be upgraded to global well-posedness [25, Theorem
5.1].
Before we state the well-posedness result, we introduce some notations which will be
used in the paper. For any µR,Mµdenotes the collection of Riemannian manifolds with
sectional curvature Kthat satisfies λ≤ K µfor some λR. In other words, Mµis the
set of Riemannian manifolds with bounded sectional curvature, where µis an upper bound
of K. Also, Br(p) := {xM:d(x, p)< r}is the open ball centred at p, of radius r, defined
for all pMand r > 0.
Denote
rw= min inj(M)
2,π
2µ,
with the convention that 1
µ=when µ0. Note that if Mis simply connected,
in addition to satisfying (M), then inj(M) = when µ0 (cf. [33, Corollary 6.9.1],
a consequence of the Cartan–Hadamard theorem). Consequently, in such case rw=.
Another remark is that by [16, Theorem IX.6.1], Brw(p) is strongly convex for any pM.
In particular, for any two points x, y Brw(p) there exists a unique length-minimizing
geodesic connecting xand y, that is entirely contained in Brw(p).
Theorem 2.1. (Global well-posedness [25, Theorem 5.1]) Let M∈ Mµfor some µR
and Ksatisfy (M) and (K), respectively. Take an initial density ρ0∈ P(U)and 0< r < rw
be such that supp(ρ0)Br(p)Ufor some open set U. Then, there exists a unique weak
solution ρin C([0,); P(U)) starting from ρ0to the interaction equation (1.1); furthermore,
supp(ρt)Br(p)for all t0.
We note that due to the attractive nature of the potential, Br(p) in Theorem 2.1 is an
invariant set for the dynamics. Also, [25, Theorem 5.1] does not have rwas the maximal
radius for well-posedness, but lists a smaller value instead. We explain in Appendix A how
a small change to the argument used there leads to the well-posedness result in Theorem
2.1.
2.2. Wasserstein distance and gradient flow formulation. We will use the intrinsic
2-Wasserstein distance to investigate the asymptotic behaviour of solutions to (1.1). For
UMopen, and ρ, σ ∈ P(U), this distance is defined as:
W2(ρ, σ) = inf
γΠ(ρ,σ)ZU×U
d(x, y)2dγ(x, y)1/2
,
where Π(ρ, σ)⊂ P(U×U) is the set of transport plans between ρand σ, i.e., the set of
elements in P(U×U) with first and second marginals ρand σ, respectively.
Denote by P2(U) the set of probability measures on Uwith finite second moment; when
Uis bounded we have P2(U) = P(U). The space (P2(U), W2) is a metric space.
摘要:

LONG-TIMEBEHAVIOUROFINTERACTIONMODELSONRIEMANNIANMANIFOLDSWITHBOUNDEDCURVATURERAZVANC.FETECAUANDHANSOLPARKAbstract.Weinvestigatethelong-timebehaviourofsolutionstoanonlocalpartialdif-ferentialequationonsmoothRiemannianmanifoldsofboundedsectionalcurvature.Theequationmodelsself-collectivebehaviourwithi...

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