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