Nonlocal-interaction equations on uniformly prox-regular sets

We study the well-posedness of a class of nonlocal-interaction equations on general domains $\Omega\subset \mathbb{R}^d$, including nonconvex ones. We show that under mild assumptions on the regularity of domains (uniform prox-regularity), for $\lambda$-geodesically convex interaction and external potentials, the nonlocal-interaction equations have unique weak measure solutions. Moreover, we show quantitative estimates on the stability of solutions which quantify the interplay of the geometry of the domain and the convexity of the energy. We use these results to investigate on which domains and for which potentials the solutions aggregate to a single point as time goes to infinity. Our approach is based on the theory of gradient flows in spaces of probability measures.


INTRODUCTION
1.1. Description of the problem. We study a continuum model of agents interacting via a potential W and subject to an external potential V confined to a closed subset Ω ⊂ R d . Such systems arise in modeling macroscopic behavior of agents interacting in geometrically confined domains. The domain boundary may be an environmental obstacle, like a river, or the ground itself, as in the models of locust patterns discussed in [3,17,18]. We consider systems in which the environmental boundaries limit the movement but not the interaction between agents. To illustrate, the the agents can still see each others over a river even if they are not able to traverse it.
We describe configurations of agents as measures supported on the given domain. This allows to study both the discrete case, when an individual agent carries a positive mass, and the continuum limit in which a system with many agents is described by a function giving the density of agents. The measure describing the agents interacting over time satisfies a nonlocal-interaction equation in the sense of weak measure solutions. The theory of weak measure solutions to nonlocal-interaction equations was developed in [2,5]. In [22] systems of interacting agents on domains with boundary were considered in a setting which allowed for heterogeneous environments, but required the sets to be convex with C 1 boundary. Here we consider general domains which are not required to be convex and whose boundary may not be differentiable. The geometrical confinement introduces a constraint on the possible velocity fields of the agents at the boundary. We consider the situation in which there is no additional friction at the boundary. More precisely, for smooth domains, the velocity of the agents at the boundary is the projection of what the velocity would be, for the given configuration if there was no boundary, to the half plane of vectors pointing inside the domain. That is, inward pointing velocities at the boundary are unchanged, while the outward pointing velocities are projected on the tangent plane to the boundary. The measure µ( · ) describing the agent configuration becomes a distributional solution of the equation where P x is the projection of the velocities to inward pointing ones. When considering domains which are not C 1 the question is what should the velocity of agents be at a boundary point where the domain is not differentiable. Similar questions have been encountered in studies of differential inclusions on moving domains (general sweeping processes), see [8,9,19] and references therein. We rely on notions developed there to properly define the cone of admissible directions at a boundary points and the proper way to project the velocity to the allowable cone. In particular we consider the equation (1.1) with projection P x defined in (1.6) (P x = P T (Ω,x) ).
While one would like to consider very general domains there are limits to possible domains on which a well-posedness of measure solutions can be developed. Namely, if the domains have an inside corner, then it is not possible for the measure solutions of (1.1) to be stable, as we discuss in Remark 1.11. It turned out that a class of domains which is rather general and allows for a well-posedness theory are the (uniformly) prox-regular domains (see Definition 1.3). Prox-regular domains are the sets which have an outside neighborhood such that for each of its points there exists a unique closest point on the boundary. In particular prox-regular domains can have outside corners and outside cusps, but not inside corners.
Our main result is the well-posedness of weak measure solutions, described in Definition 1.1, of the nonlocal-interaction equation (1.1) on uniformly prox-regular domains. To show it we rely on further structure the equation possesses. Namely to the interaction W , we associate interaction energy W (µ) = 1 2 Ω Ω W (x − y)dµ(x)dµ(y), the and to potential V , the potential energy We define the energy E by The energy E is a dissipated quantity of the evolution (1.1), and furthermore the equation can be interpreted as the gradient flow of the energy with respect to the Wasserstein metric. Our strategy is to first show the existence of the gradient flow solutions to (1.1) in the space of probability measures endowed with the Wasserstein metric. The gradient flow in the space of probability measures endowed with Wasserstein metric was first used in [11] for Fokker-Planck equations. The gradient-flow approach to well-posedness of nonlocal-interaction equations was developed in [2,5] and extended to C 1 domains with boundary in [22]. Furthermore the gradient flow approach was used to study systems in which there are state constraints that determine the set of possible velocities, in particular in crowd motion models [1,12,13] where the constraint on the L ∞ -norm of the density of agents, which leads to an L 2 -projection of velocity field. After establishing the well-posedness of gradient-flow solutions we show the well-posedness of weak measure solutions.
To show the existence of gradient flow solutions, we use particle approximations, that is we use a sequence of delta masses µ n 0 = ∑ k(n) j=1 m j δ x n j to approximate the initial data µ 0 and solve (1.1) with initial data µ n 0 . Here the notion of gradient flow solutions (and weak measure solutions) provides the advantage that we can work with delta measures, which makes the particle approximation meaningful. With discrete initial data µ n 0 , (1.1) becomes a system of ordinary differential equations, we solve the ODE system and prove that the solutions µ n ( · ) converges to some µ( · ) by establishing the stability property of solutions to (1.1) with different initial data. We then show that the limit curve µ( · ) is a gradient flow solution to (1.1) with initial data µ 0 by proving that µ( · ) achieves the maximal dissipation of the associated energy, and is thus the steepest descent of the energy.
The novelty here is that even though the domain Ω is only prox-regular (not necessarily convex or C 1 ) and the velocity field is discontinuous (due to the projection P), the ODE systems are still well-posed (refer to Theorem 2.6) and the stability of solutions µ n ( · ) in Wasserstein metric d W is valid with explicit dependence on the prox-regularity constant (refer to Proposition 3.1). Under semi-convexity assumptions on the potential functions W and V , this enables us to show the wellposedness, that is existence and stability of weak measure solutions to (1.1) in three different cases: Ω bounded and prox-regular (Theorem 1.5 and Thorem 1.6), Ω unbounded and convex (Theorem 1.9), and Ω unbounded and prox-regular with compactly supported initial data µ 0 (Theorem 1.10). We can also generalize the well-posedness results to time-dependent interaction and external po- We also give sufficient conditions on the shape of Ω to ensure the existence of an interaction potentials W such that solutions µ( · ) to (1.1) aggregate to a single delta mass as time goes to infinity (Theorem 1.13 and Remark 6.1) .
The projection P x is described below and formally defined in (1.6) (with P x = P T (Ω,x) ).
Note that the test function φ does not have to be zero on the boundary of Ω, and thus the no-flux boundary condition is imposed in a weak form.
We now define the projection P x . When ∂ Ω ∈ C 1 is smooth and oriented, the definition of P x is given in [22,23], and it is given by P is the unit outward normal vector to the boundary at x ∈ ∂ Ω. When Ω is only proxregular, to define P x , we need to recall some notations from non-smooth analysis, see [4,7], in order to replace the normal vector field and the inward and outward directions. Definition 1.2. Let S be a closed subset of R d . We define the proximal normal cone to S at x by, Note that for x ∈ S \ ∂ S, N P (S, x) = {0} and by convention for x ∈ S, N P (S, x) = / 0. The notion of normal cone extends the concept of outer normal of a smooth set in the sense that if S is a closed subset of R d with boundary ∂ S an oriented C 2 hypersurface, then for each x ∈ ∂ S, N P (S, x) = R + ν(x) where ν(x) is the unit outward normal to S at x. We now recall the notion of uniform prox-regular sets.
where B η (y) denotes the open ball centered at y with radius η > 0.
Note that an equivalent characterization, see [7,15], is given by: S is η-prox-regular if for any y ∈ S, x ∈ ∂ S and v ∈ N P (S, x), Observe that if S is closed and convex, then S is ∞-prox-regular. We now turn to the tangent cones.
Definition 1.4. Let S be a closed subset of R d and x ∈ S, define the Clarke tangent cone by and denote the Clarke normal cone by x FIGURE 1. The set S is prox-regular but not convex. At the corner point x ∈ ∂ S, the tangent and normal cones are denoted by T (S, x) and N(S, x).
Note that T C (S, x), N C (S, x) are closed convex cones, also by convention N C (S, x) = / 0 for all x ∈ S. In general, we only have N P (S, x) ⊂ N C (S, x) and the inclusion can be strict. However, for η-prox-regular set S, we have N P (S, x) = N C (S, x), see [7,15]. In that case, we put the normal cone and tangent cone as N(S, x) and T (S, x) respectively, and for any vector w ∈ R d , we define the projection onto the tangent cone by P T (S,x) (w), i.e., Since T (S, x) is a closed convex cone, the infimum is always attained, and P T (S,x) is well-defined. For notation simplicity, since the set we are considering Ω is not changing, we write P x instead of P T (Ω,x) and when the context is clear, we put P for P x . With these preliminaries, we can now state the main results of this work.
we say that f is λ -geodesically convex on a convex set S if for any x, y ∈ S we have We call f locally λ -geodesically convex if there exist a sequence of compact convex sets K n ⊂ R d and a sequence of constants λ n such that K n ⊂ K n+1 , n K n = R d and f is λ n -geodesically convex on K n . Note that f is λ -geodesically convex on a convex set S implies for any x, y ∈ S The main assumptions depend on the domain Ω and the support of initial data. In fact, we separate our results in three cases: Ω bounded, Ω unbounded and convex, and Ω unbounded with compactly supported initial data. The assumptions are very similar in nature based on the convexity of the potentials V and W and on their growth behavior at ∞ in the unbounded cases. We assume that both potentials V and W are λ V -and λ W -convex respectively, possibly locally convex. Finally, in case V and W are λ -locally convex, we can assume, without loss of generality, that V and W share the same sequence of compact convex sets, K k in the definition of locally λ -geodesic convexity, i.e., K k ⊂ K k+1 , k∈N K k = R d with V and W being λ V,k and λ W,k -geodesically convex on K k .
In case Ω is bounded, we assume that We call a locally absolutely continuous curve µ(t) ∈ P 2 (Ω) a gradient flow of the energy func- where ∂ E (µ(t)) is the subdifferential of E at µ(t) (as given in Definition 3.3) and v(t) is the tangent velocity of the curve [0, ∞) t → µ(t) ∈ P 2 (Ω) at µ(t), which we recall in Section 3.
For a locally absolutely continuous curve [0, T ] t → µ(t) ∈ P 2 (Ω) with respect to 2-Wasserstein metric d W , we denote its metric derivative by The main results of this paper is the well-posedness of weak measure solutions: existence and stability, with arbitrary initial data. We establish it using an approximation scheme and the theory of gradient flows in spaces of probability measures. Theorem 1.5. Assume Ω is bounded and satisfies (M1) and W,V satisfy (A1), (A2). Then there exists a locally absolutely continuous curve µ( · ) ∈ P 2 (Ω) such that µ( · ) is a gradient flow with respect to E . Moreover, µ( · ) is a weak measure solution to (1.1).
Observe that in the stability estimate for solutions (1.12), we find two contributions due to the λ -convexity of the potentials and the η-prox-regular property of the domain Ω respectively.
On R n when µ 1 (0) and µ 2 (0) have the same center of mass λ − W can be replaced by λ W in (1.12). Thus when the potential W is uniformly geodesically convex, λ W > 0 and thus there is exponential contraction of solutions. On bounded domains this is not the case since interaction with boundary can change the center of mass of a solution. Nevertheless part of the claim can be recovered. We consider the case that V ≡ 0. Denote the set of singletons by Ξ = {δ x : x ∈ R d }. Note that we included the singletons which are not in the set Ω, since the center of mass for measures on a non-convex Ω may lie outside the set.
The proposition implies that solution can aggregate to a point (in perhaps infinite time) even on a nonconvex domain. We ask on what domains there exists a potential for which for any initial datum this aggregation property holds. We provide a sufficient condition on the shape of Ω for aggregation to hold: Let diam(Ω) = sup x,y∈Ω |x − y|.
Note that the constant in η > 1 2 diam(Ω) cannot be improved, as the example in Remark 6.1 shows.
We generalize the two existence and stability results to the unbounded case in two different settings. In case Ω is unbounded, and for general initial data µ 0 , possibly with noncompact support, we give the global assumptions: for some constants λ W , λ V ∈ R and C > 0, The main result in this setting reads as: Assume Ω is unbounded and satisfies (GM1) and W,V satisfy (GA1)-(GA4), then for any µ 0 ∈ P 2 (Ω), there exists a gradient flow solution µ( · ) with respect to E such that µ( · ) is a weak measure solution to (1.1). Moreover, for a.e. t > 0 Similarly, if µ 1 ( · ), µ 2 ( · ) are two weak measure solutions to (1.1) with initial data µ 1 0 , µ 2 0 respectively, then for any t ≥ 0. Also the weak measure solution is characterized by the system of Evolution Variational Inequalities: for a.e. t > 0 and for all ν ∈ P 2 (Ω).
Since Ω is convex means Ω is ∞-prox-regular, the stability estimate (1.15) and EVI (1.16) in the convex setting are consistent with the estimates in the η-prox-regular setting by taking η = ∞ in (1.12) and (1.11).
The convexity assumption is needed since on nonconvex unbounded domains we do not know how to control the error due to lack of convexity (as measured by the prox-regularity (1.5)) in the stability of solutions. However, we can show that control assuming compactly supported initial data. Therefore, when Ω is unbounded and the initial data µ 0 has compact support, we assume there exist some constants η > 0, λ W , λ V ∈ R,C > 0 such that the following local assumptions hold Note that the conditions (LA1) and (LA3) are satisfied whenever V and W are C 2 functions on R d , which is the case in many practical applications. We show in this setting the following theorem about existence and stability for weak measure solutions for initial data with compact support. Theorem 1.10. Given that Ω is unbounded and satisfies (M1), and W,V satisfy (LA1)-(LA4). If where λ W,k , λ V,k are the geodesic convexity constants of W and V in K k and Ω k = Ω ∩ K k .
Let us point out that we are not able to get the system of Evolution Variational Inequalities in its whole generality although they hold for compactly supported reference measures.
The red arrows show the projected velocity field Pv on γ 1 and γ 2 , which are driving the particles apart from each other.

1.4.
Strategy of the proof. The strategy to construct weak measure solutions to (1.1) is to show the existence of gradient flow with respect to E . We approximate the initial data µ 0 in Wasserstein metric by µ n which we show based on the well-posedness theory from non-convex sweeping process differential inclusions with perturbations. For the general theory of sweeping processes we refer to [8,9,19] and references therein. To be precise, based on [9] there exists a locally absolutely continuous We then show that the solution to (1.19) is actually a solution to (1.18).
Next we explore the properties of the sequence of solutions {µ n ( · )} n . In particular, • When Ω is bounded or Ω is unbounded but convex, we first prove the stability of µ n (t) is a constant depending only on W,V . Thus µ n (t) converges to some µ(t) in P 2 (Ω) as n → ∞. Since µ n satisfies the energy dissipation inequality, by the lower semicontinuity property, we are able to show that µ( · ) also satisfies the desired energy dissipation inequality (1.21) We then show the chain rule, forṽ(t) is the tangent velocity of µ( · ) at time t which together with the energy dissipation inequality yields that µ( · ) is a gradient flow with respect to E and a weak measure solution to (1.1). • When Ω is unbounded and only η-prox-regular, we first show that the support of the solutions µ n (t) grows at most exponentially, i.e.

1.5.
Outline. The paper is organized as follows.
In Section 2, we show the properties of the projection P and then give the existence results for the discrete projected systems (1.18).
In Section 3, under the assumption that Ω is bounded, we prove the stability of solutions to the discrete projected systems µ n ( · ), i.e. (1.20). Thus µ n ( · ) converge to an absolutely continuous curve µ( · ). We show that µ( · ) is curve of maximum slope for the energy E and moreover a gradient flow solution of (1.1). We then show that µ( · ) is also a weak measure solution and that weak measure solutions satisfy the stability property (1.12). At the end of the section, we show that solutions are characterized by the system of Evolution Variational Inequalities (1.11). Section 4 addresses the case of unbounded, convex Ω and general initial data µ 0 ∈ P 2 (Ω), that is Theorem 1.9. The proof of Theorem 1.9 is similar to Theorem 1.5 and Theorem 1.6, we only concentrate on the key differences.
Section 5 is devoted to the case when Ω is unbounded and only η-prox-regular with supp(µ 0 ) compact. We show that the support of the solutions to the discrete projected systems (1.18) satisfy exponential growth condition (1.23). By similar stability results as in Section 3, µ n ( · ) still converges to a locally absolutely continuous curve µ( · ) and µ( · ) is a solution to (1.1) with the desired energy dissipation (1.21). We then give the proof of the stability result (1.17) for solutions with control on growth of supports. We end the section by making a remark about well-posedness of (1.1) with time-dependent potentials W,V .
In the last Section 6, we prove Proposition 1.7 and discuss the conditions on the shape of the domain Ω such that there exist interaction potentials W for which solutions µ( · ) of (1.1) aggregate to a singleton (a single delta mass).

EXISTENCE OF SOLUTIONS TO DISCRETE SYSTEMS
In this section, we first show properties of the projection P, in particular the lower semicontinuity and convexity property of P. Then we give the existence result of solutions to the discrete projected systems (1.18).
Recall that the tangent and normal cones T (Ω, x) and N(Ω, x) are closed convex cones by Definition 1.4.
Proposition 2.1 is a direct consequence of Moreau's decomposition theorem, see [14,16] for the proof.
Proof. We first show the lower semicontinuity property. Let {x n } n ⊂ Ω, {v n } n ⊂ R d be such that lim n→∞ x n = x ∈ Ω, lim n→∞ v n = v. If x n ∈Ω for all n sufficiently large, then P x n (v n ) = v n and we have |P x (v)| 2 ≤ |v| 2 = lim n→∞ |v n | 2 . And for any x ∈Ω, we have x n ∈Ω for n sufficiently large, thus So we only need to check for x ∈ ∂ Ω and {x n } n ⊂ ∂ Ω such that lim n→∞ x n = x. Denote the decomposition of v n as in Proposition 2.1 by and v n T , v n N = 0. For any subsequence, which we do not relabel, such that there exists w N ∈ R d and lim n→∞ v n N = w N , we claim that w N ∈ N(Ω, x) and v − w N , w N = 0. Indeed, since Ω is η-prox-regular, We turn to the convexity property. For any fixed

Convexity is verified.
We cite the following result from [8,9] about the existence of differential inclusions Theorem 2.3. Assume that S is η-prox-regular as defined in Definition 1.3 and F : R d x → F(x) ∈ R d is a continuous function with at most linear growth, i.e., there exists some constant C > 0 such that |F(x)| ≤ C(1 + |x|).
Then the differential inclusion has at least one locally absolutely continuous solution.
Note that the theorems, for example Theorem 5.1 from [9], are more general than Theorem 2.3. However, we only need the simplified version for our purpose. We also notice that (2.1) implies that Then the continuity of x(t) and the fact that S is closed imply that x(t) ∈ S for all t ≥ 0. For completeness, we give a sketch of proof here.
Proof. For T < 1 2C where C is constant in the growth condition of F. For n ∈ N, take the partition 0 = t n 0 < t n 1 < ... < t n n = T and define δ n x n i ).
Since 2CT < 1 we have uniformly in n max 0≤i≤n x n i ≤ We now define the approximation solution by Notice that x n can also be written as and Z n (t) = Z n i for t n i ≤ t < t n i+1 . We have for a.e. t ∈ [t n i ,t n i+1 ) x n (t) + Z n (t) = Π n (t) ∈ N (S, x n (t n i )) .
Since Π n (t) ≤ Z n i for t ∈ (t n i ,t n i+1 ], we know there exists a subsequence of n, which we do not relabel, such that Π n Π, Z n Z as n → ∞ weakly in L 2 [0, T ]. We then have by (2.2) that x n converges locally uniformly to x with We now claim that x(t) is a solution to the differential inclusion on [0, T ]. First we check that Then by Proposition 2.1 from [9], we know for a.e. t ∈ [0, T ], Now we only need to check that Z(t) = F(x(t)). We know that Z n (t) = F(x n (t n i ) for t n i ≤ t < t n i+1 . Defineũ n byx n (t) = x n (t n i ) for t n i ≤ t < t n i+1 and note Z n (t) = F(x n (t n i ) = F(x n (t)). Theñ x n converges locally uniformly to x. Together with the fact that F is continuous, The claim is proved.
We now show that the solutions for the differential inclusions are actually solutions for the projected systems.
Lemma 2.4. Assume that S is η-prox-regular by Definition 1.3 and x(t) is a locally absolutely continuous solution to the differential inclusion (2.1). Then Proof. Since S is η-prox-regular, it is tangentially regular, that is where T (S, x) is defined in Definition 1.4 and K(S, x) is the contingent cone defined as We refer to [4] for the details. Now note that for a.e. ṫ Thus ẋ(t), n(x(t)) = 0 for any n(x(t)) ∈ N(S, x(t)). From the differential inclusion (2.1),we know that −F(x(t)) =ẋ(t) + n(x(t)) for some n(x(t)) ∈ N(S, x(t)). Together with fact thatẋ(t) ∈ T (S, x(t)) and ẋ(t), n(x(t)) = 0, by Proposition 2.1 x(t) = P x(t) (−F (x(t))) , as claimed.
We turn to the existence of solutions to the discrete projected system (1.18), which we write as for i = 1, · · · , n. For that purpose we apply Theorem 2.3 and Lemma 2.4 for S = Ω n and . To do that, we first check that Ω n is η-prox-regular.
Proof. To see Ω n is also η-prox-regular, first it is direct that Ω n is a closed set. Now for any which implies x i ∈ P Ω x i + αv i for 1 ≤ i ≤ n. By the equivalent definition of η-prox-regularity of Ω (1.5), we then have for any y = (y 1 , · · · , y n ) ∈ Ω n . Thus Ω n is η-prox-regular by (1.5). We now turn to the relations between the normal cones. For x = (x 1 , · · · , x n ) ∈ Ω n and v = Thus N(Ω n , x) = N(Ω, x 1 ) × · · · × N(Ω, x n ).
Now we give the main result regarding the existence of solutions to projected discrete systems.
Proof. We just need to check the conditions for Theorem 2.3 to apply. We already know that Ω n is η-prox-regular. If Ω is bounded and W,V satisfy (A1)-(A2), then the mapping Ω n y = (y 1 , · · · , y n ) → F(y) = (∇W * µ(y 1 ) + ∇V (y 1 ), · · · , ∇W * µ(y n ) + ∇V (y n )) where µ = ∑ n i=1 m i δ y i , is continuous and bounded. Extend F to R dn so that F is still continuous and bounded. Then by Theorem 2.3 there exists an absolutely continuous solution to the differential inclusion Similarly, if Ω is unbounded and ∇W, ∇V satisfy liner growth conditions (GA2) and (GA4), then the mapping R dn y = (y 1 , · · · , y n ) → F(y) = (∇W * µ(y 1 ) + ∇(y 1 ), · · · , ∇W * µ(y n ) + ∇V (y n )) where µ = ∑ n i=1 m i δ y i , is continuous and has linear growth on R dn . By Theorem 2.3, we still have an absolutely continuous solution to (2.6). Now consider (2.6) in components yields for 1 ≤ i ≤ n and v i ( Then similar argument as in Lemma 2.4 gives

EXISTENCE AND STABILITY OF SOLUTIONS WITH Ω BOUNDED
In this section, we show the existence and stability of solutions to (1.1) for the case when Ω is bounded, prox-regular and W,V satisfy (A1)-(A2).
We approximate µ 0 ∈ P 2 (Ω) by µ n 0 = ∑ k(n) i=1 m n i δ x n i such that x n i ∈ Ω and lim n→∞ d W (µ 0 , µ n 0 ) = 0. By Theorem 2.6, for each n ∈ N there exists a a locally absolutely continuous solution to and µ n (t) = ∑ k(n) j=1 m n j δ x n j (t) . It is a straightforward calculation to see that for any Thus µ n (t) satisfies The following proposition contains the key estimate on the stability of solutions in the discrete case. In particular it shows how the stability in Wasserstein metric d W defined in (1.3) is affected by the lack of convexity of the domain.
Proposition 3.1. Assume that Ω is bounded and satisfies (M1), W,V satisfy (A1) and (A2). Then for two solutions µ n ( · ) and µ m ( · ) to the discrete system with different initial data µ n 0 , µ m 0 , we have for all t ≥ 0 Proof. Note that µ n ( · ) is solution to the continuity equation for v n (t, x) = −∇W * µ n (t)(x) − ∇V (x). Since the discrete solutions may have different numbers of particles we use a transportation plan to relate them. Let γ t ∈ Γ o (µ n (t), µ m (t)) be the optimal plan between µ m and µ n defined in (1.4  y)) , x − y dγ t (x, y).
We first establish the contractivity the solutions would have if the boundary conditions were not present and then account for the change due to velocity projection at the boundary. For v n , v m , by (A1) and (A2), that is the convexity of W and V , For the boundary effect, by the fact that Ω is η-prox-regular we have (1.5), thus .

Remark 3.2.
Our goal is to show that µ( · ) is a weak measure solution of (1.1). The most immediate idea would be to try to pass to limit directly in Definition 1.1. However note that since P x is not continuous in x and thus the velocity field governing the dynamics is not continuous (at the boundary of Ω). Given that µ n converge to µ only in the weak topology of measures, the lack of continuity of velocities prevents us to directly pass to limit in the integral formulation given in Definition 1.1. To show that µ( · ) is a weak measure solution of (1.1) we use the theory of gradient flows in the spaces of probability measures P 2 (Ω). Namely, we establish that µ( · ) satisfies the steepest descent property with respect to the total energy E defined in (1.2) by showing µ n ( · ) satisfies such property and the property is stable under the weak topology of measures (convergence in the Wasserstein metric d W ).
We turn to the introducing the elements of the theory of gradient flows in the space of probability measures. Definition 3.3. Let µ ∈ P 2 (Ω), a vector field ξ on Ω is said to be in the subdifferential of E at µ if ξ ∈ L 2 (µ), i.e.
For the proof of the theorem, refer to Theorem 8.3.1 from [2]. We call the unique vector field v(t) the tangent velocity field and define where v(t) is the tangent velocity field for µ(t).
Before proving the theorem, we need the following Lemma 3.7. Assume (M1) holds for Ω and ν n ∈ P 2 (Ω) converges narrowly to ν ∈ P 2 (Ω) with sup n Ω |x| 2 dν n (x) < ∞, then Proof. Similar argument as in Lemma 2.7 from [5] yields that ∇W * ν n converges weakly to ∇W * ν, i.e., for any φ Then by Proposition 2.2 we proved in Section 2 and Proposition 6.42 from [10], we know that there exist two sequences of bounded continuous functions a i , b i such that for all Taking supremum over i ∈ N and using Lebesgue's monotone convergence theorem then gives We now start to prove the theorem Proof of Theorem 3.6. We first show that the map t → E (µ n (t)) is locally absolutely continuous.

Also by Lemma 3.7, for any
By Fatou's lemma, we then have We now claim that To see that, first notice that sup n t s | (µ n ) | 2 (r)dr < ∞, so | (µ n ) | ∈ L 2 ([s,t]) and converges weakly in L 2 ([s,t]) to some function A as n → ∞. We then have for any 0

Thus we have |µ |(r) ≤ A(r)
for s ≤ r ≤ t, which then implies The claim is proved. Now take n → ∞ in (3.11)gives as desired.
Thus µ( · ) is a gradient flow with respect to E and by (3.14), a weak measure solution to (1.1).
We turn to the proof of Theorem 1.6 Proof of Theorem 1.6. We show (1.12) first. Let µ 1 ( · ), µ 2 ( · ) be two solutions to (1.1), by Theorem 8.4.7 and Lemma 4.3.4 from [2], we have For v i , by (A1) and (A2) similar argument as in the proof of Proposition 3.1 gives By the fact that Ω is η-prox-regular we have where Plugging back into (3.19) yields 1 2 Then by Gronwall's inequality we have for all t ≥ 0 (1.12) is proved. For (1.11), we have if µ( · ) is a weak measure solution to (1.1), then for any ν ∈ P 2 (Ω) and On the other hand, if µ 1 ( · ) satisfies (1.11) and µ 2 ( · ) is the solution to (1.1) such that µ 1 0 = µ 2 0 ,by Lemma 4.3.4 from [2] we get Again by Gronwall's inequality we have µ 1 (t) = µ 2 (t) for all t ≥ 0. Thus the weak measure solution is characterized by the system of evolution variational inequalities (1.11).

EXISTENCE AND STABILITY OF SOLUTIONS WITH Ω UNBOUNDED: GLOBAL CASE
In this section we prove the existence and stability of (1.1) with Ω unbounded, convex and W,V satisfying (GA1)-(GA4).
Proposition 4.1. Assume that Ω is unbounded and convex, W,V satisfy (GA1)-(GA4). Then for two solutions µ m ( · ), µ n ( · ) to the discrete system with different initial data µ m 0 , µ n 0 , we have for all t ≥ 0 (4.1) The proof is similar to the proof of Proposition 3.1 once we notice that since Ω is ∞-proxregular, by (1.5) for any x, y ∈ Ω So as n → ∞ we again know that µ n (t) converges to some µ(t) ∈ (P 2 (Ω) , d W ). Before proving that µ(t) is a curve of maximal slope, we need

Thus lim inf
n→∞ Ω Similarly, the condition C|x| 2 −V (x) is lower semicontinuous and bounded from below implies as claimed.
We estimate the growth of support of the solutions µ n ( · ) to (1.18).
Proof. Define r(t) = sup i |x n i (t)|. For fixed t > 0, assume that x n i (t) realizes R(t) i.e., r(t) = |x n i (t)|, then Thus r(t) ≤ r(0) exp(Ct) + exp(Ct) − 1 for r(0) ≤ n and C depending only on W,V , in particular independent of the number of particles k(n).
We can now show  Proof. We first check that for fixed n ∈ N, the function t → E (µ n (t)) is locally absolutely continuous. For fixed 0 ≤ s < t < ∞, by Lemma 4.3, ∇V (x) L ∞ (Ω∩B(r(t))) < ∞ and ∇W L ∞ (Ω∩B(r(t))−Ω∩B(r(t)) < ∞. Then by the same argument as in (3.10), t → E (µ(t)) is locally absolutely continuous. Together with Proposition 4.2, the proof is now identical to the proof of Theorem 3.11. We omit it here.

EXISTENCE AND STABILITY OF SOLUTIONS WITH Ω UNBOUNDED: COMPACTLY SUPPORTED INITIAL DATA CASE
In this section, we show the existence and stability results in the case when Ω is unbounded and W,V satisfy (LA1)-(LA4). The novelty is that λ -geodesic convexity of energy is only assumed locally (which is automatically satisfied if V and W are C 2 functions).
We start by giving the control the support of the solutions µ n (t) to (1.18). Notice that when approximating µ 0 by µ n 0 = ∑ k(n) i=1 m n i δ x n i , since supp(µ 0 ) ⊂ Ω ∩ B(r 0 ), we can take x n i ∈ Ω ∩ B(r 0 + 1) for all n ∈ N and 1 ≤ i ≤ k(n) such that we have So without loss of generality, we assume supp (µ n 0 ) ⊂ B(r 0 ) for all n ∈ N. Then by Lemma 4.3, supp (µ n (t)) ⊂ Ω ∩ B(r(t)) for r(t) ≤ (r 0 + 1) exp(Ct) for some C = C(W,V ) independent of n.
Proof. For any fixed 0 < T < ∞ and any 0 ≤ t ≤ T , we know that supp(µ n (t)) ⊂ B(r(T )) for all 0 ≤ t ≤ T uniformly in n. Let K k and λ W,k , λ V,k be the sequences of compact convex sets and constants such that W,V are λ W,k and λ V,k -geodesically convex on K k . Take k 0 be such that B(2r(T )) ⊂ K k for all k ≥ k 0 . Still denote γ t ∈ Γ o (µ n (t), µ m (t)) an optimal plan. Now notice that supp(µ n (t)), supp(µ m (t)) ⊂ B(r(t)) ∩ Ω ⊂ K k ∩ Ω = Ω k , thus where Ω t = Ω ∩ B(r(t)). Since v n (t, Thus as in the proof of Proposition 3.1, we have for 0 ≤ t ≤ T 1 2 By Gronwall's inequality, we have for all Thus as n → ∞, µ n (t) converges in P 2 (Ω) to some µ(t).
Theorem 5.2. µ( · ) is a curve of maximal slope, for any 0 ≤ s < t < ∞ Proof. We use similar argument as in Theorem 3.6 and Theorem 4.4. For any fixed n ∈ N, since supp(µ n (t)) ⊂ Ω ∩ B(r(t)), we can still control the L ∞ -norm of ∇V and ∇W . Then the same argument as in the proof of Theorem 4.4 shows that t → E (µ(t)) is locally absolutely continuous. Thus the fact that µ n are solutions to the discrete systems implies, s Ω |P x (v n (r, x)) | 2 dµ n (r, x)dr.
By Lemma 3.7 and notice that ∇W * µ n (r) + ∇V still converges weakly to ∇W * µ(r) + ∇V for any 0 ≤ r ≤ T , then Now by the same argument as in the proof of (3.13), we again obtain lim inf n→∞ t s | (µ n ) | 2 (r)dr ≥ t s |µ | 2 (r)dr.
We now start to prove Theorem 1.10 Proof of Theorem 1.10. Since µ( · ) is locally absolutely continuous, we know that there exists a unique Borel vector fieldṽ such that ∂ t µ(t) + div (µ(t)ṽ(t)) = 0 holds in the sense of distributions. For a fixed T > 0 and any µ, ν ∈ P 2 (Ω) with supp(µ), supp(ν) ⊂ B(r(T )), let γ ∈ Γ o (µ, ν). Since W,V are λ W,k and λ V,k -geodesically convex on K k ⊃ B(r(t)) ∩ Ω, we have that the function f we defined in (3.17) by taking λ = λ k is non-decreasing in t for any (x 1 , y 1 ), (x 2 , y 2 ) ∈ supp γ. Thus we still have For any 0 < t < T , and h > 0 such that Also E (µ(t)) is locally absolutely continuous, so for a.e. t > 0 which again implies Combine with (5.1) yieldsṽ and for any 0 For the contraction property (1.17), we notice that for any 0 ≤ t ≤ T < ∞ and k ∈ N such that B(r(T )) ⊂ K k where Ω k = Ω ∩ K k . Thus by Gronwall's inequality, we have for all 0 ≤ t ≤ T Remark 5.3. When the external and interaction potentials are time-dependent V = V (t, x),W = W (t, x), then with some modifications of the arguments we have before, we can still show the existence and stability results of the solutions to (1.1) in all the three different cases as in the timeindependent settings before. For example, we assume that there are constants λ ∈ R, η > 0 and a positive function β ∈ L 1 ([0, ∞)) such that (M1) Ω is bounded and η-prox-regular.

AGGREGATION ON NONCONVEX DOMAINS
In this section, we consider the following question: what are the conditions on Ω to ensure the existence of an interaction potential W such that the solution µ( · ) to (1.1) aggregates to a singleton (delta mass) as time goes to infinity?
Let Ω be bounded and η-prox-regular, V ≡ 0 and W satisfy (A1) for some λ W > 0, such that Theorem 1.5 holds and we have a weak measure solution µ( · ) to (1.1). We recall Ξ = {δ x : x ∈ R d } the set of singletons, and start to estimate the evolution of d W (µ( · ), Ξ), the distance of µ( · ) to Ξ. That is we prove Proposition 1.7.
Proof. It suffices to show that for all t ≥ 0 since then by Gronwall's inequality the result follows. By shifting time we can assume that t = 0. Denote the center of mass for µ 0 byx, that is x = Ω xdµ(0, x). It is direct computation to show that d W (µ(0), Ξ) = d W (µ(0), δx) , and for any t > 0, d W (µ(t), Ξ) ≤ d W (µ(t), δx). Thus Now we follow similar argument as in the proof of Proposition 3.1. To be precise, by (3.5) with µ n (t) = µ(t), µ m (t) ≡ δx, we have Combine the estimates yields We now prove Theorem 1.8.
Proof. It turns out that the quadratic interaction potential leads to the sharpest bound for general domains. Furthermore, since multiplying a potential by a positive constant only leads to a constant rescaling in time of the dynamics, we consider W (x) = 1 2 |x| 2 . To verify the inequality (1.13) note that ∇W (x) = x, HessW (x) = Id and λ W = 1. Thus sup Ω−Ω |∇W | ≤ sup x,y∈Ω |x − y| = diam(Ω) and Remark 6.1. We notice that (1.13) implies that lim t→∞ d W µ(t), δx (t) = 0 wherex(t) = Ω xdµ(t, x) is the center of mass for µ(t). Hence as t → ∞, µ(t) converges in d W to a singleton, i.e., all mass aggregates to one point to form a delta mass of size 1. Thus Theorem 1.8 gives a sufficient condition on the shape of the domain Ω on which there exists a radially symmetric interaction potential W so that solutions aggregate to a point. We note that the simple condition given in the theorem is also sharp in the following sense: for any ε > 0 there exists Ω bounded and η-prox-regular with 0 < η ≤ ( 1 2 − ε) diam(Ω), and an initial condition µ 0 such that the solution starting from µ 0 does not aggregate to a point.
Ω v(x 1 ) v(x 2 ) ε x 2 x 1 FIGURE 3. The velocity v at x 1 and x 2 are shown as the red arrows, which lie in the normal cones of the points respectively.