Lp Theory for the Multidimensional Aggregation Equation

(ii) If α ≥ 2 then the aggregation equation is globally well posed in P 2 (R d ) ∩ L p (R d ) for every p > 1.As a consequence we have existence and uniqueness for all potentials which are less singular than theNewtonian potential K(x) = |x| 2−d at the origin. In two dimensions this includes potentials with cuspsuch as K(x) = |x| 1/2 . In three dimensions this includes potentials that blow up such as K(x) = |x| −1/2 .From [5, 16] we know that the support of compactly supported solutions shrinks to a point in finite time,proving the second assertion in point (i) above. The first part of (i) and statement (ii) are direct corollaryof Theorem 1, Definition 2 and the fact that α ≥ 2 implies ∆K bounded.In the case where α = 1, i.e. K(x) ∼ |x| as |x| → 0, the previous Theorem gives local well posedness inP 2 (R d ) ∩ L p (R d ) for all p > p s =dd−1. The next Theorem shows that it is not possible to obtain local wellposedness in P 2 (R d ) ∩ L p (R d ) for p < p s =dd−1 .Theorem 4 (Critical p-exponent to generate instantaneous mass concentration). Suppose K(x) = |x| in aneighborhood of the origin, and suppose ∇K is compactly supported (or decays exponentially fast at infinity).Then, for any p < p s =dd−1 , there exists initial data in P 2(R d ) ∩ L p (R d ) for which a delta Dirac appearsinstantaneously in the measure solution.In order to make sense of the statement of the previous Theorem, we need a concept of measure solution.The potentials K(x) = |x| is semi-convex, i.e. there exist λ ∈ R such that K(x) − λ 2 |x|2 is convex. In[16], Carrillo et al. prove global well-posedness in P 2 (R d ) of the aggregation equation with semi-convexpotentials. The solutions in [16] are weak measure solutions - they are not necessarily absolutely continuouswith respect to the Lebesgue measure. Theorems 3 and 4 give a sharp condition on the initial data in orderfor the solution to stay absolutely continuous with respect to the Lebesgue measure for short time. Theorem4 is proven in Section 4.Finally, in section 5 we consider a class of potential that will be referred to as the class of naturalpotentials. A potential is said to be natural if it satisfies thata) it is a radially symmetric potential, i.e.: K(x) = k(|x|),b) it is smooth away from the origin and it’s singularity at the origin is not worse than Lipschitz,c) it doesn’t exhibit pathological oscillation at the origin,d) its derivatives decay fast enough at infinity.All these conditions will be more rigorously stated later. It will be shown that the gradient of naturalpotentials automatically belongs to W 1,q for q < d, therefore, using the results from the sections 2 and 3,we have local existence and uniqueness in P 2 (R d ) ∩ L p (R d ), p >dd−1 .A natural potential is said to be repulsive in the short range if it has a local maximum at the originand it is said to be attractive in the short range if it has a local minimum at the origin. If the maximum(respectively minimum) is strict, the natural potential is said to be strictly repulsive (respectively strictlyattractive) at the origin. The main theorem of section 5 is the following:Theorem 5 (Osgood condition for global well posedness). Suppose K is a natural potential.(i) If K is repulsive in the short range, then the aggregation equation is globally well posed in P 2 (R d ) ∩L p (R d ), p > d/(d − 1).(ii) If K is strictly attractive in the short range, the aggregation equation is globally well posed in P 2 (R d ) ∩L p (R d ), p > d/(d − 1), if and only ifr ↦→ 1k ′ (r)is not integrable at 0. (1.9)3

By globally well posed in P 2 (R d ) ∩ L p (R d ), we mean that for any initial data in P 2 (R d ) ∩ L p (R d ) theunique solution of the aggregation equation will exist for all time and will stay in P 2 (R d ) ∩ L p (R d ) for alltime.Notice that the exponent d/(d − 1) is not sharp in this theorem.Condition (1.9) will be refered as the Osgood condition. It is easy to understand why the Osgoodcondition is relevant while studying blowup: the quantityT (d) =∫ d0drk ′ (r)can be thought as the amount of time it takes for a particle obeying the ODE Ẋ = −∇K(X) to reach theorigin if it starts at a distance d from it. For a potential satisfying the Osgood condition, T (d) = +∞, whichmeans that the particle can not reach the origin in finite time. The Osgood condition was already shownin [5] to be necessary and sufficient for global well posedness of L ∞ -solutions. Extension to L p requires L pestimates rather than L ∞ estimates. See also [43] for an example of the use of the Osgood condition in thecontext of the Euler equations for incompressible fluid.The “only if” part of statement (ii) was proven in [5] and [16]. In these two works it was shown thatif (1.9) is not satisfied, then compactly supported solutions will collapse into a point mass – and thereforeleave L p – in finite time. In section 5 we prove statement (i) and the ’if’ part of statement (ii).2 Existence of L p -solutionsIn this section we show that if the interaction potential satisfies∇K ∈ W 1,q (R d ), 1 < q < +∞, (2.10)and if the initial data is nonnegative and belongs to L p (R d ) (p and q are Hölder conjugates) then there existsa solution to the aggregation equation. Moreover, either this solution exists for all times, or its L p -normblows up in finite time. The duality between L p and L q guarantees enough smoothness in the velocity fieldv = −∇K ∗ u to define characteristics. We use the characteristics to construct a solution. The argumentis inspired by the existence of L ∞ solutions of the incompressible 2D Euler equations by Yudovich [44] andof L ∞ solutions of the aggregation equation [4]. Section 3 proves uniqueness provided u ∈ P 2 . We prove inTheorem 18 of the present section that if u 0 ∈ P 2 then the solution stays in P 2 . Finally we prove that if inaddition to (2.10), we haveess sup ∆K < +∞, (2.11)then the solution constructed exists for all time.Most of the section is devoted to the proof of the following theorem:Theorem 6 (Local existence). Consider 1 < q < ∞ and p its Hölder conjugate. Suppose ∇K ∈ W 1,q (R d )and suppose u 0 ∈ L p (R d ) is nonnegative. Then there exists a time T ∗ > 0 and a nonnegative functionu ∈ C([0, T ∗ ], L p (R d )) ∩ C 1 ([0, T ∗ ], W −1,p (R d )) such thatu ′ (t) + div ( u(t) v(t) ) = 0 ∀t ∈ [0, T ∗ ], (2.12)Moreover the function t → ‖u(t)‖ p Lis differentiable and satisfiesp∫ddt {‖u(t)‖p L p} = −(p − 1)The choice of the spacev(t) = −u(t) ∗ ∇K ∀t ∈ [0, T ∗ ], (2.13)u(0) = u 0 . (2.14)R d u(t, x) p div v(t, x) dx ∀t ∈ [0, T ∗ ]. (2.15)Y p := C([0, T ∗ ], L p (R d )) ∩ C 1 ([0, T ∗ ], W −1,p (R d ))is motivated by the fact that, if u ∈ Y p and ∇K ∈ W 1,q , then the velocity field is automatically C 1 in spaceand time:4

Lemma 7. Consider 1 < q < ∞ and p its Hölder conjugate. If ∇K ∈ W 1,q (R d ) and u ∈ Y p thenu ∗ ∇K ∈ C 1 ( [0, T ∗ ] × R d) and ‖u ∗ ∇K‖ C1 ([0,T ∗ ]×R d ) ≤ ‖∇K‖ W 1,q (R d ) ‖u‖ Y p(2.16)where the norm ‖·‖ C 1 ([0,T ∗ ]×R d ) and ‖·‖ Y pare defined by∣ ∣ ∣ ∣∣∣‖v‖ C1 ([0,T ∗ ]×R d ) = sup∂vd∑ ∣∣∣|v| + sup[0,T ∗ ]×R d [0,T ∗ ]×R ∂t ∣ + ∂v ∣∣∣sup , (2.17)d [0,T ∗ ]×R ∂x d i‖u‖ Yp= sup ‖u(t)‖ Lp (R d ) +t∈[0,T ∗ ]supt∈[0,T ∗ ]i=1‖u ′ (t)‖ W −1,p (R d ) . (2.18)Proof. Recall that the convolution between a L p -function and a L q -function is continuous and sup x∈R d |f ∗g(x)| ≤ ‖f‖ L p ‖g‖ L q Therefore, since ∇K and ∇K xi are in L q , the mappingf ↦→ ∇K ∗ fis a bounded linear transformation from L p (R d ) to C 1 (R d ), where C 1 (R d ) is endowed with the norm‖f‖ C 1= supx∈R d |f(x)| +d∑sup | ∂f (x)|.x∈R ∂x d iSince u ∈ C([0, T ∗ ], L p ) it is then clear that u ∗ ∇K ∈ C([0, T ∗ ], C 1 ). In particular w(t, x) = (u(t) ∗ ∇K) (x)and ∂w∂x i(t, x) are continuous on [0, T ∗ ] × R d . Let us now show that ∂w∂t(t, x) exists and is continuous on[0, T ∗ ] × R d . Since u ′ (t) ∈ C([0, T ∗ ], W −1,p ) and ∇K ∈ W 1,q , we havei=1∂w∂t (t, x) = − (u′ (t) ∗ ∇K) (x) = −〈u ′ (t), τ x ∇K〉where 〈 , 〉 denote the pairing between the two dual spaces W −1,p (R d ) and W 1,q (R d ), and τ x denote thetranslation by x. Since x ↦→ τ x ∇K is a continuous mapping from R d to W 1,q it is clear that ∂w∂t(t, x) iscontinuous with respect to space. The continuity with respect to time come from the continuity of u ′ (t) withrespect to time. Inequality (2.16) is easily obtained.Remark 8. Let us point out that (2.12) indeed makes sense, when understood as an equality in W −1,p .Since v ∈ C([0, T ∗ ], C 1 (R d )) one can easily check that uv ∈ C([0, T ∗ ], L p (R d )). Also recall that the injectioni : L p (R d ) → W −1,p (R d ) and the differentiation ∂ xi : L p (R d ) → W −1,p (R d ) are bounded linear operators.Therefore it is clear that both u and div(uv) belong to C([0, T ∗ ], W −1,p (R d )). Equation (2.12) has to beunderstand as an equality in W −1,p .The rest of this section is organized as follows. First we give the basic a priori estimates in subsection 2.1.Then, in subsection 2.2, we consider a mollified and cutted-off version of the aggregation equation for whichwe have global existence of smooth and compactly supported solutions. In subsection 2.3 we show that thecharacteristics of this approximate problem are uniformly Lipschitz continuous on [0, T ∗ ]×R d , where T ∗ > 0is some finite time depending on ‖u 0 ‖ L p. In subsection 2.4 we pass to the limit in C([0, T ∗ ], L p ). To do thiswe need the uniform Lipschitz bound on the characteristics together with the fact that the translation byx, x ↦→ τ x u 0 , is a continuous mapping from R d to L p (R d ). In subsection 2.5 we prove three theorems. Wefirst prove continuation of solutions. We then prove that L p -solutions which start in P 2 stay in P 2 as longas they exist. And finally we prove global existence in the case where ∆K is bounded from above.2.1 A priori estimatesSuppose u ∈ Cc 1 ((0, T ) × R d ) is a nonnegative function which satisfies (2.12)-(2.13) in the classical sense.Suppose also that K ∈ Cc∞ (R d ). Integrating by part, we obtain that for any p ∈ (1, +∞):∫∫du(t, x) p dx = −(p − 1) u(t, x) p div v(t, x)dx ∀t ∈ (0, T ). (2.19)dt R d R d5

As a consequence we have:ddt ‖u(t)‖p L ≤ (p − 1)‖div v(t)‖ p L ∞‖u(t)‖p Lp ∀t ∈ (0, T ), (2.20)and by Hölder’s inequality:ddt ‖u(t)‖p L ≤ (p − 1)‖∆K‖ pL q‖u(t)‖p+1 L . (2.21)pWe now derive L ∞ estimates for the velocity field v = −∇K ∗ u and its derivatives. Hölder’s inequalityeasily givesSince ∂v∂t∂u= −∇K ∗∂t ∣|v(t, x)| ≤ ‖u(t)‖ L p‖∇K‖ L q ∀(t, x) ∈ (0, T ) × R d , (2.22)∥ ∂v j∣ (t, x)∂x i∣ ≤ ‖u(t)‖ L p∂ 2 K ∥∥∥L∥∀(t, x) ∈ (0, T ) × R d . (2.23)∂x i dx j q= ∇K ∗ div(uv) = ∆K ∗ uv we have∣∂v ∣∣∣∂t (t, x) ≤ ‖u(t)v(t)‖ L p‖∆K‖ L q ≤ ‖u(t)‖ L p‖v(t)‖ L ∞‖∆K‖ L q,which in light of (2.22) gives∣ ∂v ∣∣∣∣ ∂t (t, x) ≤ ‖u(t)‖ 2 L p‖∇K‖ L q‖∆K‖ L q ∀(t, x) ∈ (0, T ) × Rd . (2.24)2.2 Approximate smooth compactly supported solutionsIn this section we deal with a smooth version of equation (2.12)-(2.14). Suppose u 0 ∈ L p (R d ), 1 < p < +∞,and ∇K ∈ W 1,q (R d ). Consider the approximate problemu t + div(uv) = 0 in (0, +∞) × R d , (2.25)v = −∇K ɛ ∗ u in (0, +∞) × R d , (2.26)u(0) = u ɛ 0, (2.27)where K ɛ = J ɛ K, u ɛ 0 = J ɛ u 0 and J ɛ is an operator which mollifies and cuts-off, J ɛ f = (fM Rɛ ) ∗ η ɛ whereη ɛ (x) is a standard mollifier:η ɛ (x) = 1 ( x)∫ɛ d η , η ∈ Cc ∞ (R d ), η ≥ 0, η(x)dx = 1,ɛR dand M Rɛ (x) is a standard cut-off function: M Rɛ (x) = M( x R ɛ), R ɛ → ∞ as ɛ → 0,M ∈ C ∞ c (R d ),Let τ x denote the translation by x, i.e.:⎧⎨⎩M(x) = 1 if |x| ≤ 1,0 < M(x) < 1 if 1 < |x| < 2,M = 0 if 2 ≤ |x|.τ x f(y) := f(y − x).It is well known that given a fixed f ∈ L r (R d ), 1 < r < +∞, the mapping x ↦→ τ x f from R d to L r (R d )is uniformly continuous. In (iv) of the next lemma we show a slightly stronger result which will be neededlater.Lemma 9 (Properties of J ɛ ). Suppose f ∈ L r (R d ), 1 < r < +∞, then6

(i) J ɛ f ∈ C ∞ c (R d ),(ii) ‖J ɛ f‖ L r ≤ ‖f‖ L r,(iii) lim ɛ→0 ‖J ɛ f − f‖ L r = 0,(iv) The family of mappings x ↦→ τ x J ɛ f from R d to L r (R d ) is equicontinuous, i.e.: for each δ > 0, there isa η > 0 independent of ɛ such that ‖τ x J ɛ f − τ y J ɛ f‖ L r ≤ δ if |x − y| ≤ η.Proof. Statements (i) and (ii) are obvious. If f is compactly supported, one can easily prove (iii) by notingthat fM Rɛ = f for ɛ small enough. If f is not compactly supported, (iii) is obtained by approximating f bya compactly supported function and by using (ii). Let us now turn to the proof of (iv). Using (ii) we obtain‖τ x J ɛ f − J ɛ f‖ L r ≤ ‖(τ x M Rɛ )(τ x f) − M Rɛ f)‖ L r≤ ‖τ x M Rɛ − M Rɛ ‖ L ∞‖τ x f‖ L r + ‖M Rɛ ‖ L ∞‖τ x f − f‖ L r.Because x ↦→ τ x f is continuous, the second term can be made as small as we want by choosing |x| smallenough. Since ‖τ x M Rɛ − M Rɛ ‖ L ∞ ≤ ‖∇M Rɛ ‖ L ∞|x| ≤ 1 R ɛ‖∇M‖ L ∞|x|, the first term can be made as smallas we want by choosing |x| small enough and independently of ɛ.Proposition 10 (Global existence of smooth compactly-supported approximates). Given ɛ, T > 0, thereexists a nonnegative function u ∈ C 1 c ((0, T ) × R d ) which satisfy (2.27) in the classical sense.Proof. Since u ɛ 0 and K ɛ belong to Cc∞function u ɛ satisfying(R d ), we can use theorem 3 p. 1961 of [28] to get the existence of au ɛ ∈ L ∞ (0, T ; H k ), u ɛ t ∈ L ∞ , (0, T ; H k−1 ) for all k, (2.28)u ɛ t + div (u ɛ (−∇K ɛ ∗ u ɛ )) = 0 in (0, T ) × R d , (2.29)u ɛ (0) = u ɛ 0, (2.30)u ɛ (t, x) ≥ 0 for a.e. (t, x) ∈ (0, T ), ×R d . (2.31)Statement (2.28) implies that u ɛ ∈ C((0, T ); H k−1 ). Using the continuous embedding H k−1 (R d ) ⊂ C 1 (R d )for k large enough we find that u ɛ and u ɛ x i, 1 ≤ i ≤ d, are continuous on (0, T ) × R d . Finally, (2.29) showsthat u ɛ t is also continuous on (0, T ) × R d . We have proven that u ɛ ∈ C 1 ((0, T ) × R d ). It is then obvious thatv ɛ = −∇K ɛ ∗ u ɛ ∈ C 1 ((0, T ) × R d ). Note moreover that|v ɛ (x, t)| ≤ ‖u ɛ ‖ L∞ (0,T ;L 2 )‖∇K ɛ ‖ L 2for all (t, x) ∈ (0, T ) × R d . This combined with the fact that v ɛ is in C 1 shows that the characteristics arewell defined and propagate with finite speed. This proves that u ɛ is compactly supported in (0, T ) × R d(because u ɛ 0 is compactly supported in R d ).2.3 Study of the velocity field and the induced flow mapNote that K ɛ and u ɛ are in the right function spaces so that we can apply to them to the a priori estimatesderived in section 2.1. In particular we have:ddt ‖uɛ (t)‖ L p ≤ (p − 1)‖∆K‖ L q‖u ɛ (t)‖ p+1L p ,‖u ɛ (0)‖ L p ≤ ‖u 0 ‖ L p.Using Gronwall inequality and the estimate on the supremum norm of the derivatives derived in section 2.1we obtain:7

Lemma 11 (uniform bound for the smooth approximates). There exists a time T ∗ > 0 and a constantC > 0, both independent of ɛ, such that‖u ɛ (t)‖ L p ≤ C ∀t ∈ [0, T ∗ ], (2.32)|v ɛ t, x)|, |v ɛ x i(t, x)|, |v ɛ t(t, x)| ≤ C ∀(t, x) ∈ [0, T ∗ ] × R d . (2.33)From (2.33) it is clear that the family {v ɛ } is uniformly Lipschitz on [0, T ∗ ] × R d , with Lipschitz constantC. We can therefore use the Arzela-Ascoli Theorem to obtain the existence of a continuous function v(t, x)such thatv ɛ → v uniformly on compact subset of [0, T ∗ ] × R d . (2.34)It is easy to check that this function v is also Lipschitz continuous with Lipschitz constant C. The Lipschitzand bounded vector field v ɛ generates a flow map X ɛ (t, α), t ∈ [0, T ∗ ], α ∈ R d :∂X ɛ (t, α)= v ɛ (X ɛ (t, α), t),∂tX ɛ (0, α) = α,where we denote by X t ɛ : R d → R d the mapping α ↦→ X ɛ (t, α) and by X −tɛ the inverse of X t ɛ.The uniform Lipschitz bound on the vector field implies uniform Lipschitz bound on the flow map andits inverse (see for example [4] for a proof of this statement) we therefore have:Lemma 12 (uniform Lipschitz bound on Xɛ t and Xɛ−t ). There exists a constant C > 0 independent of ɛsuch that:(i) for all t ∈ [0, T ∗ ] and for all x 1 , x 2 ∈ R d|Xɛ(x t 1 ) − Xɛ(x t 2 )| ≤ C|x 1 − x 2 | and |Xɛ−t (x 1 ) − x −tɛ (x 2 )| ≤ C|x 1 − x 2 |,(ii) for all t 1 , t 2 ∈ [0, T ∗ ] and for all x ∈ R d|X t1ɛ (x) − X t2ɛ (x)| ≤ C|t 1 , t 2 | and |Xɛ−t1(x) − X −t2ɛ (x)| ≤ C|t 1 − t 2 |.The Arzela-Ascoli Theorem then implies that there exists mapping X t and X −t such thatX t ɛ k(x) → X t (x) uniformly on compact subset of [0, T ∗ ] × R d ,X −tɛ k(x) → X −t (x) uniformly on compact subset of [0, T ∗ ] × R d .Moreover it is easy to check that X t and X −t inherit the Lipschitz bounds of X t ɛ and X −tɛ .Since the mapping X t : R d → R d is Lipschitz continuous, by Rademecher’s Theorem it is differentiablealmost everywhere. Therefore it makes sense to consider its Jacobian matrix DX t (α). Because of Lemma12-(i) we know that there exists a constant C independent of t and ε such thatsupαɛR d |det DX t (α)| ≤ C and supα∈R d |det DX t ɛ(α)| ≤ C.By the change of variable we then easily obtain the following Lemma:Lemma 13. The mappings f ↦→ f ◦ X −t and f ↦→ f ◦ Xɛ−t , t ∈ [0, T ∗ ], ɛ > 0, are bounded linear operatorsfrom L p (R d ) to L p (R d ). Moreover there exists a constant C ∗ independent of t and ɛ such that‖f ◦ X −t ‖ L p ≤ C ∗ ‖f‖ L p and ‖f ◦ X −tɛ ‖ L p ≤ C ∗ ‖f‖ L p for all f ∈ L p (R d ).8

Note that Lemma 12-(ii) impliesand thereforeThis gives us the following lemma:Lemma 14. let Ω be a compact subset of R d , thenwhere the compact set Ω + Ct is defined by2.4 Convergence in C([0, T ∗ ), L p )|X t ɛ(α) − α| ≤ Ct for all (t, α) ∈ [0, T ∗ ) × R d ,|X t (α) − α| ≤ Ct for all (t, α) ∈ [0, T ∗ ) × R d .X t ɛ(Ω) ⊂ Ω + Ct and X t (Ω) ⊂ Ω + Ct,Ω + Ct := {x ∈ R d : dist (x, Ω) ≤ Ct}.Since u ɛ and v ɛ = u ɛ ∗ ∇K are C 1 functions which satisfyu ɛ t + v ɛ · ∇u ɛ = −(div v ɛ )u ɛ and u ɛ (0) = u ɛ 0 (2.35)we have the simple representation formula for u ɛ (t, x), t ∈ [0, T ∗ ), x ∈ R d :u ɛ (t, x) = u ɛ 0(Xɛ−t (x))e − R tdiv 0 vɛ (s,X −(t−s)ɛ(x))ds = u ɛ 0(X −t (x)) a ɛ (t, x).Lemma 15. There exists a function a(t, x) ∈ C 1 ([0, T ∗ ) × R d ) and a sequence ɛ k → 0 such thata ɛ k(t, x) → a(t, x) uniformly on compact subset of [0, T ∗ ] × R d . (2.36)Proof. By the Arzela-Ascoli Theorem, it is enough to show that the familyb ɛ (t, x) :=∫ t0div v ɛ (s, Xɛ−(t−s) (x))dxis equicontinuous and uniformly bounded. The uniform boundedness simply come from the fact that|div v ɛ | = |u ɛ ∗ ∆K ɛ | ≤ ‖u ɛ ‖ L p‖∆K‖ L q.Let us now prove equicontinuity in space, i.e., we want to prove that for each δ > 0, there is η > 0 independentof ɛ and t such that|b ɛ (t, x 1 ) − b ɛ (t, x 2 )| ≤ δ if |x 1 − x 2 | ≤ η.First, note that by Hölder’s inequality we haveɛ|b ɛ (t, x 1 ) − b ɛ (t, x 2 )| ≤∫ t0‖u ɛ (s)‖ L p‖τ ξ J ɛ ∆K − τ ζ J ɛ ∆K‖ L qdswhere ξ stands for Xɛ−(t−s) (x 1 ) and ζ for Xɛ−(t−s) (x 2 ). Then equicontinuity in space is a consequence ofLemma 9 (iv) together with the fact thatwhere C is independent of t, s and ɛ.|Xɛ−(t−s) (x 1 ) − Xɛ −(t−s) (x 2 )| ≤ C|x 1 − x 2 |9

Let us finally prove equicontinuity in time. First not that, assuming that t 1 < t 2 ,b ɛ (t 1 , x) − b ɛ (t 2 , x) =−∫ t10∫ t2t 1div v ɛ (s, X −(t1−s) (x)) − div v ɛ (s, X −(t2−s) (x))dsdiv v ɛ (s, Xɛ−(t2−s) (x))ds.Since div v ɛ is uniformly bounded we clearly have∫ t2∣ div v ɛ (s, Xɛ−(t2−s) (x))ds∣ ≤ C|t 1 − t 2 |.t 1The other term can be treated exactly as before, when we proved equicontinuity in space.Recall that the functionu ɛ (t, x) = u ɛ 0(Xɛ −t (x)) a ɛ (t, x)satisfies the ɛ-problem (2.27). We also have the following convergences:Define the functionu ɛ 0 → u 0 in L p (R d ), (2.37)Xɛ −tk(x) → X −t (x) unif. on compact subset of [0, T ∗ ] × R d , (2.38)a ɛ k(t, x) → a(t, x) unif. on compact subset of [0, T ∗ ] × R d , (2.39)v ɛ k(t, x) → v(t, x) unif. on compact subset of [0, T ∗ ] × R d . (2.40)u(t, x) := u 0 (X −t (x)) a(t, x). (2.41)Convergence (2.37)-(2.40) together with Lemma 13 and 14 allow us to prove the following proposition.Proposition 16. : u, u ɛ , uv and u ɛ v ɛ all belong to the space C([0, T ∗ ), L p (R d )). Moreover we have:Proof. Straight forward, see appendix at the end of the paper.We now turn to the proof of the main theorem of this section.u ɛ k→ u in C([0, T ∗ ), L p ), (2.42)u ɛ kv ɛ k→ uv in C([0, T ∗ ), L p ). (2.43)Proof of Theorem 6. Let φ ∈ C ∞ c(0, T ∗ ) be a scalar test function. It is obvious that u ɛ and v ɛ satisfy:∫ T∗∫ T∗− u ɛ (t) φ ′ (t) dt + div ( u ɛ (t) v ɛ (t) ) φ(t) dt = 0, (2.44)00v ɛ (t, x) = (u ɛ (t) ∗ ∇K ɛ )(x) for all (t, x) ∈ [0, T ∗ ] × R d , (2.45)u ɛ (0) = u ɛ 0, (2.46)where the integrals in (2.44) are the integral of a continuous function from [0, T ∗ ] to the Banach spaceW −1,p (R d ). Recall that the injection i : L p (R d ) → W −1,p (R d ) and the differentiation ∂ xi : L p (R d ) →W −1,p (R d ) are bounded linear operator. Therefore (2.42) and (2.43) implyu ɛ k→ u in C([0, T ∗ ], W −1,p (R d )),div [u ɛ kv ɛ k] → div [uv] in C([0, T ∗ ], W −1,p (R d )), (2.47)hich is more than enough to pass to the limit in relation (2.44). To pass to the limit in (2.45), it is enoughto note that for all (t, x) ∈ [0, T ∗ ] × R d we have|(u ɛ (t) ∗ ∇K ɛ ) − (u(t) ∗ ∇K)(x)| ≤ ‖u ɛ (t) − u(t)‖ L p‖∇K ɛ ‖ L q + ‖u(t)‖ L p‖∇K ɛ − ∇K‖ L q, (2.48)10

and finally it is trivial to pass to the limit in relation (2.46).Equation (2.44) means that the continuous function u(t) (continuous function with values in W −1,p (R d ))satisfies (2.12) in the distributional sense. But (2.12) implies that the distributional derivative u ′ (t) is itselfa continuous function with value in W −1,p (R d ). Therefore u(t) is differentiable in the classical sense, i.e, itbelongs to C 1 ([0, T ∗ ], W −1,p (R d )), and (2.12) is satisfied in the classical sense.We now turn to the proof of (2.15). The u ɛ ’s satisfies (2.19). Integrating over [0, t], t < T ∗ , we get∫ t ∫‖u ɛ (t)‖ p L = p ‖uɛ 0‖ p L − (p − 1) u ɛ (s, x) p div v ɛ (s, x) dxdt. (2.49)p0 R dProposition 16 together with the general inequality()‖|f| p − |g| p ‖ L 1 ≤ 2p ‖f‖ p−1L + p ‖g‖p−1 L‖f − g‖ p L p (2.50)implies that(u ɛ ) p , u p ∈ C([0, T ∗ ], L 1 (R d )), (2.51)(u ɛ k) p → u p ∈ C([0, T ∗ ], L 1 (R d )). (2.52)On the other hand, replacing ∇K by ∆K in (2.48) we see right away thatCombining (2.51)-(2.54) we obtainSo we can pass to the limit in (2.49) to obtaindiv v, div v ɛ ∈ C([0, T ∗ ], L ∞ (R d )), (2.53)div v ɛ k→ div v in C([0, T ∗ ], L ∞ (R d )). (2.54)u p div v, (u ɛ ) p div v ɛ ∈ C([0, T ∗ ], L 1 (R d )), (2.55)(u ɛ k) p div v ɛ k→ u p div v in C([0, T ∗ ], L 1 (R d )). (2.56)∫ t ∫‖u(t)‖ p L = ‖u 0‖ p p L − (p − 1) u(s, x) p div v(s, x) dxdt.p0 R dBut (2.55) implies that the function t → ∫ u(t, x) p div v(t, x) dx is continuous, therefore the functionR dt → ‖u(t)‖ p Lp is differentiable and satisfies (2.15).2.5 Continuation and conserved propertiesTheorem 17 (Continuation of solutions). The solution provided by Theorem 6 can be continued up to atime T max ∈ (0, +∞]. If T max < +∞, then lim t→Tmax sup τ∈[0,t] ‖u(τ)‖ L p = +∞Proof. The proof is standard. One just needs to use the continuity of the solution with respect to time.Theorem 18 (Conservation of mass/ second moment). (i) Under the assumption of Theorem 6, and if weassume moreover that u 0 ∈ L 1 (R d ), then the solution u belongs to C([0, T ∗ ], L 1 (R d )) and satisfies ‖u(t)‖ L 1 =‖u 0 ‖ L 1 for all t ∈ [0, T ∗ ].(ii) Under the assumption of Theorem 6, and if we assume moreover that u 0 has bounded second moment,then the second moment of u(t) stays bounded for all t ∈ [0, T ∗ ].Proof. We just need to revisit the proof of Proposition 16. Since u 0 ∈ L 1 ∩ L p it is clear thatu ɛ 0 = J ɛ u 0 → u 0 in L 1 (R d ) ∩ L p (R d ). (2.57)11

Using convergences (2.57), (2.38), (2.39) and (2.40) we prove thatu ɛ k→ u in C([0, T ∗ ], L 1 ∩ L p ). (2.58)The proof is exactly the same than the one of Proposition 16 . Since the aggregation equation is a conservationlaw, it is obvious that the smooth approximates satisfy ‖u ɛ (t)‖ L 1 = ‖u ɛ 0‖ L 1. Using (2.58) we obtain‖u(t)‖ L 1 = ‖u 0 ‖ L 1.We now turn to the proof of (ii). Since the smooth approximates u ɛ have compact support, their secondmoment is clearly finite, and the following manipulation are justified:∫∫(∫) 1/2 (∫) 1/2d|x| 2 u ɛ (t, x)dx = 2 ⃗x · v ɛ du ɛ (x) ≤ 2 |x| 2 u ɛ (t, x)dx |v ɛ | 2 u ɛ (t, x)dxdt R d R d R d R(∫) d 1/2≤ C |x| 2 u ɛ (t, x)dx . (2.59)R dAssume now that the second moment of u 0 is bounded. A simple computation shows that if η ɛ is radiallysymmetric, then |x| 2 ∗ η ɛ = |x| 2 + second moment of η ɛ . Therefore∫∫|x| 2 u ɛ 0(x)dx ≤ |x| 2 ∗ η ɛ (x) u 0 (x)dxR d R∫d ∫≤ |x| 2 u 0 (x)dx + |x| 2 η ɛ (x)dxR d R∫d≤ |x| 2 u 0 (x)dx + 1 for ɛ small enough. (2.60)R dInequality (2.60) come from the fact that the second moment of η ɛ goes to 0 as ɛ goes to 0. Estimate (2.59)together with (2.60) provide us with a uniform bound of the second moment of the u ɛ (t) which only dependsthe second moment of u 0 . Since u ɛ converges to u in L 1 , we obviously have, for a given R and t:∫∫|x|≤R|x| 2 u(t, x)dx = limɛ→0∫|x|≤R|x| 2 u ɛ (t, x)dx ≤ lim supɛ→0R d |x| 2 u ɛ (t, x)dx.Since R is arbitrary, this show that the second moment of u(t, ·) is bounded for all t for which the solutionexists.Combining Theorem 17 and 18 together with equality (2.15) we get:Theorem 19 (Global existence when ∆K is bounded from above). Under the assumption of Theorem 6,and if we assume moreover that u 0 ∈ L 1 (R d ) and ess sup ∆K < +∞, then the solution u exists for all times(i.e.: T max = +∞).Proof. Equality (2.15) can be written∫ddt {‖u(t)‖p L p} = (p − 1)R d u(t, x) p (u(t) ∗ ∆K)(x) dx. (2.61)Since ∆K is bounded from above we have(u(s) ∗ ∆K)(x) ≤ (ess sup ∆K)∫u(s, x)dx = (ess sup ∆K) ‖u 0 ‖ L 1 .R d (2.62)Combining (2.61), (2.62) and Gronwall inequality gives‖u(t)‖ p L p ≤ ‖u 0‖ p L p e(p−1)(ess sup ∆K)‖u0‖ L 1 t ,so the L p -norm can not blow-up in finite time which, because of Theorem 17, implies global existence.12

3 Uniqueness of solutions in P 2 (R d ) ∩ L p (R d )In this section we use an optimal transport argument to prove uniqueness of solutions in P 2 (R d ) ∩ L p (R d ),when ∇K ∈ W 1,q for 1 < q < ∞ and p its Hölder conjugate. To do that, we shall follow the steps in [19],where the authors extend the work of Loeper [34] to prove, among other, uniqueness of P 2 ∩ L ∞ -solutionsof the aggregation equation when the interaction potential K has a Lipschitz singularity at the origin.One can easily check that solutions of the aggregation equation constructed in previous sections aredistribution solutions, i.e. they satisfy∫ T ∫ ( )∫∂ϕ0 R ∂t (t, x) + v(t, x) · ∇ϕ(t, x) u(t, x) dx dt = ϕ(0, x)u 0 (x) dx (3.63)N R Nfor all ϕ ∈ C0 ∞ ([0, T ∗ ) × R N )). A function u(t, x) satisfying (3.63) is said to be a distribution solution to thecontinuity equation (2.12) with the given velocity field v(t, x) and initial data u 0 (x). In fact, it is uniquelycharacterized by∫∫u(t, x) dx = u 0 (x) dxBX −t (B)for all measurable set B ⊂ R d , see [1]. Here X t : R d → R d is the flow map associated with the velocityfield v(t, x) and X −t is its inverse. In the optimal transport terminology this is equivalent to say that X ttransports the measure u 0 onto u(t) (u(t) = X t #u 0 ).We recall, for the sake of completeness, [19, Theorem 2.4], where several results of [2, 35, 22, 1] are puttogether.Theorem 20 ([19]). Let ρ 1 and ρ 2 be two probability measures on R N , such that they are absolutely continuouswith respect to the Lebesgue measure and W 2 (ρ 1 , ρ 2 ) < ∞, and let ρ θ be an interpolation measurebetween ρ 1 and ρ 2 , defined as in [34] byρ θ = ((θ − 1)T + (2 − θ)I R N ) # ρ 1 (3.64)for θ ∈ [1, 2], where T is the optimal transport map between ρ 1 and ρ 2 due to Brenier’s theorem [12] and I R Nis the identity map. Then there exists a vector field ν θ ∈ L 2 (R N , ρ θ dx) such thatdi.dθ ρ θ + div(ρ θ ν θ ) = 0 for all θ ∈ [1, 2].∫ii. ρ θ |ν θ | 2 dx = W2 2 (ρ 1 , ρ 2 ) for all θ ∈ [1, 2].R Niii. We have the L p -interpolation estimate‖ρ θ ‖ L p (R N ) ≤ max { }‖ρ 1 ‖ L p (R N ), ‖ρ 2 ‖ L p (R N )for all θ ∈ [1, 2].Here, W 2 (f, g) is the Euclidean Wasserstein distance between two probability measures f, g ∈ P(R n ),{∫∫} 1/2W 2 (f, g) = inf|v − x| 2 dΠ(v, x) , (3.65)Π∈Γ R n ×R nwhere Π runs over the set of joint probability measures on R n × R n with marginals f and g.Now we are ready to prove the uniqueness of solutions to the aggregation equation.Theorem 21 (Uniqueness). Let u 1 , u 2 be two bounded solutions of equation (2.12) in the interval [0, T ∗ ]with initial data u 0 ∈ P 2 (R d ) ∩ L p (R d ), 1 < p < ∞ and assume that v is given by v = −∇K ∗ u, with Ksuch that ∇K ∈ W 1,q (R N ), p and q conjugates. Then u 1 (t) = u 2 (t) for all 0 ≤ t ≤ T ∗ .13

Here ∂ 0 K is the unique element of minimal norm in the subdifferential of K. Simply speaking, sinceK(x) = |x| is smooth away from the origin and radially symmetric, we have ∂ 0 K(x) = x|x|for x ≠ 0 and∂ 0 K(0) = 0, and thus:∫(∂ 0 x − yK ∗ µ)(x) =dµ(y). (4.77)|x − y|Note that, µ t being a measure, it is important for ∂ 0 K to be defined for every x ∈ R d so that (4.75) makessense. Equation (4.74) means that∫ +∞0∫y≠xR d ( dψdt (x, t) + ∇ψ(x, t) · v t(x))dµ t (x) dt = 0, (4.78)for all ψ ∈ C ∞ 0 (R d × (0, +∞)). From (4.77) it is clear that |v t (x)| ≤ 1 for all x and t, therefore the aboveintegral makes sense.The main Theorem of this section is the following:Theorem 24 (Instantaneous mass concentration). Consider the initial datau 0 (x) ={L|x| d−1+ɛ if |x| < 1,0 otherwise,(4.79)where ɛ ∈ (0, 1) and L :=( ∫|x| 0 we haveµ t ({0}) > 0,i.e., mass is concentrated at the origin instantaneously and the solution is no longer continuous with respectto the Lebesgue measure.Theorem 24 is a consequence of the following estimate on the velocity field:Proposition 25. Let (µ t ) t∈(0,+∞) be the unique measure solution of the aggregation equation with interactionpotential K(x) = |x| and with initial data (4.79). Then, for all t ∈ [0, +∞) the velocity field v t = −∂ 0 K ∗ µ tis focussing and there exists a constant C > 0 such that|v t (x)| ≥ C|x| 1−ɛ for all t ∈ [0, +∞) and x ∈ B(0, 1). (4.80)By focussing, we mean that the velocity field points inward, i.e.λ t : [0, +∞) → [0, +∞) such that v t (x) = −λ t (|x|) ⃗x x .there exists a nonnegative function4.1 Representation formula for radially symmetric measure solutionsIn this section, we show that for radially symmetric measure solutions, the characteristics are well defined.As a consequence, the solution to (2.12) can be expressed as the push forward of the initial data by the flowmap associated with the ODE defining the characteristics.In the following the unit sphere {x ∈ R d , |x| = 1} is denoted by S d and its surface area by ω d .Definition 26. If µ ∈ P(R d ) is a radially symmetric probability measure, then we define ˆµ ∈ P([0, +∞)) byfor all I ∈ B([0, +∞)).ˆµ(I) = µ({x ∈ R d : |x| ∈ I})16

Remark 27. If a measure µ is radially symmetric, then µ({x}) = 0 for all x ≠ 0, and therefore∫∇K(x − y) dµ(y) =R d \{x}∫∇K(x − y) dµ(y)R d for all x ≠ 0.In other words, for x ≠ 0, (∇K ∗ µ)(x) is well defined despite the fact that ∇K is not defined at x = 0. Asa consequence (∂ 0 K ∗ µ)(x) = (∇K ∗ µ)(x) if x ≠ 0 and (∂ 0 K ∗ µ)(0) = 0.Remark 28. If the radially symmetric measure µ is continuous with respect to the Lebesgue measure andhas radially symmetric density u(x) = ũ(|x|), then ˆµ is also continuous with respect to the Lebesgue measureand has density û, whereû(r) = ω d r d−1 ũ(r). (4.81)Lemma 29 (Polar coordinate formula for the convolution). Suppose µ ∈ P(R d ) is radially symmetric. LetK(x) = |x|, then for all x ≠ 0 we have:(∫ +∞( ) ) |x|x(µ ∗ ∇K) (x) = φ dˆµ(ρ)(4.82)ρ|x|0where the function φ : [0, +∞) → [−1, 1] is defined byφ(r) = 1 ∫re 1 − yω d |re 1 − y| · e 1dσ(y). (4.83)Proof. This come from simple algebraic manipulations. These manipulations are shown in [5].In the next Lemma we state properties of the function φ defined in (4.83).Lemma 30 (Properties of the function φ).S d(i) φ is continuous and non-decreasing on [0, +∞). Moreover φ(0) = 0, and lim r→∞ φ(r) = 1.(ii) φ(r) is O(r) as r → 0. To be more precise:Proof. Consider the function F : [0, +∞) × S d → [−1, 1] defined byφ(r)lim = 1 − 1 (y · e 1 )r→0 r ω dr>0∫S 2 dσ(y). (4.84)d(r, y) ↦→ re 1 − y|re 1 − y| · e 1. (4.85)Since F is bounded, we have thatφ(r) = 1 ∫F (r, y)dσ(y) = 1 F (r, y)dσ(y).ω d S dω d∫S d \{e 1}If y ∈ S d \{e 1 } then the function r ↦→ F (r, y) is continuous on [0, +∞) and C ∞ on (0, +∞). An explicitcomputation shows then that∂F 1 − F (r, y)2(r, y) = ≥ 0, (4.86)∂r |re 1 − y|thus φ is non-decreasing and, by the Lebesgue dominated convergence, it is easy to see that φ is continuous,φ(0) = 0 and lim r→∞ φ(r) = 1, which prove (i).To prove (ii), note that the function ∂F∂r(r, y) can be extended by continuity on [0, +∞). Therefore theright derivative with respect to r of F (r, y) is well defined:F (r, y) − F (0, y) 1 − F (0, y)2lim= = 1 − (y · e 1 ) 2 .r→0 r|y|r>017

and since ∂F/∂r is bounded on (0, +∞) × S d \{e 1 }, we can now use the Lebesgue dominated convergencetheorem to conclude:φ(r)limr→0 rr>0φ(r) − φ(0)= limr→0 rr>0= limr→0r>0∫1ω dF (r, y) − F (0, y)S dr(Remark 31. Note that for r > 0 the function ρ ↦→ φrρdσ(y) = 1 ω d∫S d1 − (y · e 1 ) 2 dσ(y).)is non increasing and continuous. Indeed, it isequal to 1 when ρ = 0 and it decreases to 0 as ρ → ∞. In particular, the integral in (4.82) is well definedfor any probability measure ˆµ ∈ P([0, +∞)).Remark 32. In dimension two, it is easy to check that lim r→1 φ ′ (r) = +∞ which implies that the derivativeof the function φ has a singularity at r = 1 and thus, that the function φ is not C 1 .Proposition 33 (Characteristic ODE). Let K(x) = |x| and let (µ t ) t∈[0,+∞) be a weakly continuous familyof radially symmetric probability measures. Then the velocity field{−(∇K ∗ µ t )(x) if x ≠ 0v(x, t) =(4.87)0 if x = 0is continuous on R d \{0} × [0 + ∞). Moreover, for every x ∈ R d , there exists an absolutely continuousfunction t → X t (x), t ∈ [0, +∞), which satisfiesddt X t(x) = v(X t (x), t) for a.e. t ∈ (0, +∞), (4.88)X 0 (x) = x. (4.89)Proof. From formula (4.82), Remark 31, and the weak continuity of the family (µ t ) t∈[0,+∞) , we obtaincontinuity in time. The continuity in space simply comes from the continuity and boundedness of thefunction φ together with the Lebesgue dominated convergence theorem.Since v is continuous on R d \{0} × [0 + ∞) we know from the Peano theorem that given x ∈ R d \{0}, theinitial value problem (4.88)-(4.89) has a C 1 solution at least for short time. We want to see that it is definedfor all time. For that, note that by a continuation argument, the interval given by Peano theorem can beextended as long as the solution stays in R d \{0}. Then, if we denote by T x the maximum time so that thesolution exists in [0, T x ) we have that either T x = ∞ and we are done, or T x < +∞, in which case clearlylim t→Tx X t (x) = 0, and we can extend the function X t (x) on [0, +∞) by setting X t (x) := 0 for t ≥ T x .The function t → X t (x) that we have just constructed is continuous on [0, +∞), C 1 on [0, +∞)\{T x }and satisfies (4.88) on [0, +∞)\{T x }. If x = 0, we obviously let X t (x) = 0 for all t ≥ 0.Finally, we present the representation formula, by which we express the solution to (2.12) as a pushforwardof the initial data. See [1] or [41] for a definition of the push-forward of a measure by a map.Proposition 34 (Representation formula). Let (µ t ) t∈[0,+∞) be a radially symmetric measure solution of theaggregation equation with interaction potential K(x) = |x|, and let X t : R d → R d be defined by (4.87), (4.88)and (4.89). Then for all t ≥ 0,µ t = X t #µ 0 .Proof. In this proof, we follow arguments from [1]. Since for a given x the function t ↦→ X t (x) is continuous,one can easily prove, using the Lebesgue dominated convergence theorem, that t ↦→ X t #µ 0 is weaklycontinuous. Let us now prove that µ t := X t #µ 0 satisfies (4.78) for all ψ ∈ C ∞ 0 (R d × (0, ∞)). Giventhat the test function ψ is compactly supported, there exist T > 0 such that ψ(x, t) = 0 for all t ≥ T . Wetherefore have:∫∫()0 = ψ(x, T )dµ T (x) − ψ(x, 0)dµ 0 (x) = ψ(X T (x), T ) − ψ(x, 0) dµ 0 (x). (4.90)R d R∫R d d18

If we now take into account that from Proposition 33 the mapping t → φ(X t (x), t) is absolutely continuous,we can rewrite (4.90) as0 ====∫R d ∫ T0∫ T ∫R d∫ T0∫ T00( ddt ψ(X t(x), t))dt dµ 0 (x) (4.91)(∇ψ(X t (x), t) · v(X t (x), t) + dψ∫R d (∇ψ(X t (x), t) · v(X t (x), t) + dψdt (X t(x), t)dt (X t(x), t)(∇ψ(x, t) · v(x, t) +∫R dψ )dt (x, t) dµ t (x) dt.d)dt dµ 0 (x) (4.92))dµ 0 (x) dt (4.93)The step from (4.92) to (4.93) holds because of the fact that |v(x, t)| ≤ 1, which justifies the use of theFubini Theorem.Remark 35 (Representation formula in polar coordinates). Let µ t and X t be as in the previous proposition.Let R t : [0, +∞) → [0, +∞) be the function such that |X t (x)| = R t (|x|). Thenˆµ t = R t #ˆµ 0 . (4.94)Remark 36. Since φ is nonnegative (Lemma 30), from (4.82), (4.87) and (4.88) we see that the functiont ↦→ |X t (x)| = R t (|x|) is non increasing.4.2 Proof of Proposition 25 and Theorem 24We are now ready to prove the estimate on the velocity field and the instantaneous concentration result. Westart by giving a frozen in time estimate of the velocity field.Lemma 37. Let K(x) = |x|, and let u 0 (x) be defined by (4.79) for some ɛ ∈ (0, 1). Then there exist aconstant C > 0 such that|(u 0 ∗ ∇K) (x)| ≥ C|x| 1−ɛ (4.95)for all x ∈ B(0, 1)\{0}.Proof. Note that if we do the change of variable s = |x|ρ|(u 0 ∗ ∇K) (x)| = |x|in equation (4.82), we find that∫ +∞0φ(s) û 0 ( |x|s )ds s 2 . (4.96)On the other hand, using (4.81) we see that the û 0 (r) corresponding to the u 0 (x) defined by (4.79) isû 0 (r) ={ωdr ɛ if r < 1,0 otherwise.(4.97)Then, plugging (4.97) in (4.96) we obtain that for all x ≠ 0|(u 0 ∗ ∇K) (x)| = ω d |x| 1−ɛ ∫ +∞|x|φ(ρ)dρ. (4.98)ρ2−ɛ In light of statement (ii) of Lemma 30, we see that the previous integral converges as |x| → 0.|(u 0 ∗ ∇K) (x)| is O(|x| 1−ɛ ) as |x| → 0 and (4.95) follows.HenceFinally, the last piece we need in order to prove Proposition 25 from the previous lemma, is the followingcomparison principle:19

Lemma 38 (Temporal monotonicity of the velocity). Let (µ t ) t∈(0,+∞) be a radially symmetric measuresolution of the aggregation equation with interaction potential K(x) = |x| . Then, for every x ∈ R d \{0} thefunctiont ↦→ |(∇K ∗ µ t )(x)|is non decreasing.Proof. Combining (4.82) and (4.94) we see that|(µ t ∗ ∇K)(x)| =(∫ +∞0)φ( |x|R t (ρ) ) dˆµ 0(ρ) . (4.99)Now, by Lemma 30, φ is non decreasing and due to Remark 35, t ↦→ R t (ρ) is non increasing. Henceforth itis clear that (4.99) is itself non decreasing.At this point, Proposition 25 follows as a simple consequence of the frozen in time estimate (4.95) togetherwith Lemma 38, and we can give an easy proof for the main result we introduced at the beginning of thesection.Proof of Theorem 24. Using the representation formula (Proposition 34) and the definition of the push forwardwe getµ t ({0}) = (X t #µ 0 )({0}) = µ 0 (Xt −1 ({0})).Then, note that the solution of the ODE ṙ = −r 1−ɛ reaches zero in finite time. Therefore, from Proposition25 and 33 we obtain that for all t > 0, there exists δ > 0 such thatX t (x) = 0 for all |x| < δ.In other words, for all t > 0, there exists δ > 0 such that B(0, δ) ⊂ Xt−1 ({0}). Clearly, given our choice ofinitial condition, we have that µ 0 (B(0, δ)) > 0 if δ > 0, and therefore µ t ({0}) > 0 if t > 0.4.3 Remark about the initial data 1/ |x| d−1An interesting open problem is wether or not there is instantaneous mass concentration when the initial datais defined by (4.79) with ɛ = 0. We right now can not answer this question, but below are some interestingremarks about this case.Lemma 39. Let u 0 be defined by (4.79) with ɛ = 0. Then there exists constants C > 0 and β > 1 such that|(∇K ∗ u 0 )(x)| ≤ C|x||x|∣log β ∣ for all x ∈ B(0, 1). (4.100)Proof. From Lemma 30 it is clear that there exists a constant α > 0 such that φ(r)/r ≤ α for all r ∈ (0, 1).We can then use (4.98) with ɛ = 0 to obtain that, for |x| < 1,( ∫ 1 ∫ )φ(ρ) dρ ∞|(u 0 ∗ ∇K)(x)| = ω d |x|ρ ρ + φ(ρ)1 ρ 2 dρ≤ ω d |x|(α|x|∫ 1|x|dρρ + cst )= ω d |x| (−α log |x| + cst) = −C|x| log |x|β ,for some constants C > 0 and β > 1.Consider the pushforward S t (r) defined by the ODEddt S t(r) = −CS t (r)∣ log S t(r)β ∣ ,S 0 (r) = r.20

This ODE has an explicit solution: for r < 1 we have( ) eCtrS t (r) = ββThe push forward of u 0 by the map S t can also be explicitly computed:(S t #u 0 )(r) ={ L(t)r α(t)if r < β 1−exp(Ct)0 otherwisewhere α(t) = (d − 1) + ∫ eCt − 1β1−exp(Ct)e Ct , and L(t) =r −α(t) dr.0In dimension 2 and greater, α(t) ≤ d − 1 for t > 0, so S t #u 0 is less singular than u 0 .5 Osgood condition for global well-posednessThis section considers the global well-posedness of the aggregation equation in P 2 (R d ) ∩ L p (R d ), dependingon the potential K. We start by giving a precise definitions of “natural potential”, “repulsive in the shortrange” and “strictly attractive in the short range”, and then we prove Theorem 5.Definition 40. A natural potential is a radially symmetric potential K(x) = k(|x|), where k : (0, +∞) → Ris a smooth function which satisfies the following conditions:(C1)sup |k ′ (r)| < +∞,r∈(0,∞)(C2) ∃ α > d such that k ′ (r) and k ′′ (r) are O(1/r α ) as r → +∞,(MN1) ∃ δ 1 > 0 such that k ′′ (r) is monotonic (either increasing or decreasing) in (0, δ 1 ),(MN2) ∃ δ 2 > 0 such that rk ′′ (r) is monotonic (either increasing or decreasing) in (0, δ 2 ).Remark 41. Note that monotonicity condition (MN1) implies that k ′ (r) and k(r) are also monotonic insome (different) neighborhood of the origin (0, δ). Also, note that (C1) and (MN1) imply(C3)limr→0 k′ (r) exists and is finite.+Remark 42. The far field condition (C2) can be dropped when the data has compact support.Definition 43. A natural potential is said to be repulsive in the short range if there exists an interval (0, δ)on which k(r) is decreasing. A natural potential is said to be strictly attractive in the short range if thereexists an interval (0, δ) on which k(r) is strictly increasing.We would like to remark that the two monotonicity conditions are not very restrictive as, in order toviolate them, a potential would have to exhibit some pathological behavior around the origin, like oscillatingfaster and faster as r → 0.5.1 Properties of natural potentialsAs a last step before proving Theorem 5, let us point out some properties of natural potentials, which showthe reason behind the choice of this kind of potentials to work with.Lemma 44. If K(x) = k(|x|) is a natural potential, then k ′′ (r) = o(1/r) as r → 0.21

Proof. First, note that since k(r) is smooth away from 0 we havek ′ (1) − k ′ (ɛ) =∫ 1ɛk ′′ (r)dr.Now, because of (MN1) we know that there exists a neighborhood of zero in which k ′′ doesn’t change sign.Therefore, letting ɛ → 0 and using (C3) we conclude that k ′′ is integrable around the origin. A simpleintegration by part, together with (C3) gives then that∫ r0k ′′ (s)s ds = −Dividing both sides by r and letting r → 0 we obtain1limr→0 r∫ r0∫ rwhich, combined with (MN2) implies lim r→0 k ′′ (r)r = 0.0k ′ (s)dr + k ′ (r)r.k ′′ (s)s ds = 0.The next lemma shows that the existence theory developed in the previous section applies to this classof potentials.Lemma 45. If K is a natural potential then ∇K ∈ W 1,q (R d ) for all 1 ≤ q < d. As a consequence, thecritical exponents p s and q s associated to a natural potential satisfyProof. Recall that∇K(x) = k ′ (|x|) x|x|andq s ≥ d and p s ≤ dd − 1 .∂ 2 ()K(x) = k ′′ (|x|) − k′ (|x|) xi x j∂x i ∂x j |x| |x| 2 + δ ijk ′ (|x|),|x|where δ ij is the Kronecker delta symbol. In order to prove the lemma, it is enough to show thatk ′ (|x|), k ′′ (|x|) and k′ (|x|)|x|belong to L q (R d ) for all 1 ≤ q < d. To do that, observe that the decay condition(C2) implies that they belong to L q (B(0, 1) c ) for all q ≥ 1. Then, we take into account that (C3)implies that k′ (r)r= O(1/r) as r → 0 and that we have seen in the previous Lemma that k ′′ (r) = o(1/r) asr → 0. This is enough to conclude, since the function x ↦→ 1/|x| is in L q (B(0, 1)) for all 1 ≤ q < d.The following Lemma together with Theorem 19 gives global existence of solutions for natural potentialswhich are repulsive in the short range, whence part (i) of Theorem 5 follows.Lemma 46. Suppose K is a natural potential which is repulsive in the short range. Then ∆K is boundedfrom above.Proof. We will prove that there is a neighborhood of zero on which ∆K ≤ 0. This combined with the decaycondition (C2) give the desired result. First, recall that ∆K(x) = k ′′ (|x|) + (d − 1)k ′ (|x|)|x| −1 . Then, sincek is repulsive in the short range, there exists a neighborhood of zero in which k ′ ≤ 0. Now, we have twopossibilities: on one hand, if lim r→0 + k ′ (r) = 0, then given r ∈ (0, +∞) there exists s ∈ (0, r) such thatk ′ (r)r= k ′′ (s). Together with (MN1), this implies that k ′′ is also non-positive in some neighborhood of zero.On the other hand if lim r→0 + k ′ (r) < 0, then the fact that k ′′ (r) = o(1/r) implies that rk′′ (r)+(d−1)k ′ (r)risnegative for r small enough.Finally, the next Lemma will be needed to prove global existence for natural potentials which are strictlyattractive in the short range and satisfy the Osgood criteria.22

Lemma 47. Suppose that K is a natural potential which is strictly attractive in the short range and satisfiesthe Osgood criteria (1.9). If moreover sup x≠0 ∆K(x) = +∞ then the following holds(Z1) lim r→0 + k ′′ (r) = +∞ and lim r→0 +k′ (r)r= +∞,(Z2) ∃δ 1 > 0 such that k ′′ (r) and k′ (r)rare decreasing for r ∈ (0, δ 1 ),(Z3) ∃δ 2 > 0 such that k ′′ (r) ≤ k′ (r)rfor r ∈ (0, δ 2 ).Proof. Let us start by proving by contradiction thatIf we suppose thatlimsupr→0 + p∈(0,r)k ′ (r)rk ′ (r)r= +∞. (5.101)< C ∀r ∈ (0, 1], (5.102)then given a sequence r n → 0 + there will exist another sequence s n → 0 + , 0 < s n < r n , such thatk ′ (r n )r n= k ′′ (s n ) < C.Since k ′′ (r) is monotonic around zero, this implies that k ′′ is bounded from above, and combining thiswith (5.102) we see that ∆K must also be bounded from above, which contradicts our assumption. Now,statements (Z1), (Z2), (Z3) follow easily: First note that if lim r→0 + k ′ (r) > 0, then clearly the Osgoodcondition (1.9) is not satisfied, whence lim r→0 + k ′ (r) = 0. This implies that for all r > 0 there existss ∈ (0, r) such thatk ′ (r)r= k ′′ (s). (5.103)Combining (5.103), (5.101) and the monotonicity of k ′′ we get that lim r→0 + k ′′ (r) = +∞ and k ′′ is decreasingon some interval (0, δ), which corresponds with the first part of (Z1) and (Z2). Now, going back to (5.103)we see that if 0 < s < r < δ then k′ (r)rddr= k ′′ (s) ≥ k ′′ (r) which proves (Z3). This implies{ k ′ }(r)= 1 r r()k ′′ (r) − k′ (r)≤ 0rand therefore k ′ (r)/r decreases on (0, δ). Thus, (5.101) implies lim r→0 + k′ (r)rcomplete.= +∞ and the proof is5.2 Global bound of the L p -norm using Osgood criteriaWe have already proven global existence when the potential is bounded from above. In this section we provethe following proposition, which allows us to prove global existence for potentials which are attractive in theshort range, satisfy the Osgood criteria, and whose Laplacian is not bounded from above. From it, secondpart of Theorem 5 follows readily. This extends prior work on L ∞ -solutions [5] to the L p case.Proposition 48. Suppose that K is a natural potential which is strictly attractive in the short range, satisfiesthe Osgood criteria (1.9) and whose Laplacian is not bounded from above (i.e. sup x≠0 ∆K(x) = +∞). Letu(t) be the unique solution of the aggregation equation starting with initial data u 0 ∈ P 2 (R d ) ∩ L p (R d ), wherep > d/(d − 1). Define the length scale( ) q/d ‖u(t)‖L 1R(t) =.‖u(t)‖ L p23

Then there exists positive constants δ and C such that the inequalityholds for all time t for which R(t) ≤ δ.dRdt ≥ −C k′ (R) (5.104)Remark 49. R(t) is the natural length scale associated with blow up of L p norm. Given that mass isconserved, R(t) > 0 means that the L p norm is bounded. The differential inequality (5.104) tell us that R(t)decays slower than the solutions of the ODE ẏ = −Ck ′ (y). Since k ′ (r) satisfies the Osgood criteria (1.9),solutions of this ODE do not go to zero in finite time. R(t) therefore stays away from zero for all time whichprovides us with a global upper bound for ‖u(t)‖ L p.Proof. Equality (2.15) can be writtenUsing the chain rule we getddt∫ddt {‖u(t)‖p L p} = (p − 1){ }‖u(t)‖ − q dL= − p(p − 1)qpd≥ −R d u(t, x) p (u(t) ∗ ∆K)(x) dx.‖u(t)‖ − q d −pL p ∫R d u(t, x) p (u(t) ∗ ∆K)(x) dx (5.105)(p − 1)q‖u(t)‖ − q dLpdsup {(u(t) ∗ ∆K)(x)} . (5.106)px∈R dTo obtain (5.104), we now need to carefully estimate sup x∈R d {(u(t) ∗ ∆K)(x)}:Lemma 50 (potential theory estimate). Suppose that K(x) = k(|x|) satisfies (C2), (Z1), (Z2) and (Z3).Suppose also that p > . Then there exists positive constants δ and C such that inequalitydd−1k ′ (R)sup (u ∗ ∆K) (x) ≤ C‖u‖ L 1x∈R Rd( ) q/d ‖u‖L 1where R =(5.107)‖u‖ L pholds for all nonnegative u ∈ L 1 (R d ) ∩ L p (R d ) satisfying R ≤ δ.Remark: By Lemma 47, potentials satisfying the conditions of Proposition 48 automatically satisfy (C2),(Z1), (Z2) and (Z3).Proof of Lemma 50. Recall that ∆K(x) = k ′′ (|x|) + (d − 1) k′ (|x|)|x|so that (Z3) implies that ∆K(x)

To go from (5.108) to (5.109) we have used the fact that k ′ is nonnegative in a neighborhood of zero (becauselim r→0 + k ′ (r)/r = +∞ ) and that u is also nonnegative. To go from (5.111) to (5.112) we have used thefact that k ′ is increasing in a neighborhood of zero (because lim r→0 + k ′′ (r) = +∞). Finally, to go from(5.112) to (5.113) we have used the fact that, since q < d, ∫ ɛ∫0 rd−1−q dr = ɛ d−q /(d − q). Let us now estimate|y|≥ɛ u(x − y)∆K(y)dy. On one hand k′′ (r) and k ′ (r)/r go to 0 as r → +∞. On the other hand k ′′ (r) andk ′ (r)/r go monotonically to +∞ as r → 0 + . Therefore for ɛ small enough we have{}sup ∆K(x) = sup k ′′ (r) + (d − 1) k′ (r)|x|≥ɛr≥ɛrwhich gives, since u is nonnegative,∫|y|≥ɛu(x − y)∆K(y)dy ≤ d ‖u‖ L 1≤ k ′′ (ɛ) + (d − 1) k′ (ɛ)ɛCombining (5.113) and (5.114) we see that for ɛ small enough we have≤ d k′ (ɛ)ɛk ′ (ɛ). (5.114)ɛsup (u ∗ ∆K) (x) ≤ c k′ (ɛ)(ɛ d/q ‖u‖ L p + ‖u‖ L 1) (5.115)x∈R ɛd(‖u‖L 1) q/d.where c is a positive constant depending on d and q. To conclude the proof, choose ɛ = R =‖u‖ L pCombining (5.106) and (5.107) allows us to get (5.104), which concludes the proof of Proposition 48.6 ConclusionsThis paper develops refined analysis for well-posedness of the multidimensional aggregation equation forinitial data in L p . An additional assumption of bounded second moment is needed as a decay condition foruniqueness; fortunately this condition is preserved by the dynamics of the equation. The results connectrecent theory developed for L ∞ initial data [4, 5, 6] to recent theory for measure solutions [16]. It turnsout that L p spaces provide a good understanding of the transition from a regular (bounded) solution to ameasure solution, which was proved to occur in [5] whenever the potential violates the Osgood condition.In [5], it is shown that for the special case of K(x) = |x|, no ‘first kind’ similarity solutions exist to describethe blowup to a mass concentration for odd space dimensions larger than one. A subsequent numerical studyof blowup [26], for K = |x|, illustrates that for dimensions larger than two, there is a ‘second kind’ self-similarblowup in which no mass concentration occurs at the initial blowup time. Rather, the solution is in L p forsome p < p s = d/(d − 1). The solution has asymptotic structure like the example constructed in Section 4of this paper, thus we expect after the initial blowup time, that it concentrates mass in a delta. We remarkthat these results provide an interesting connection to classical results for Burgers equation. Our equation inone space dimension, with K = |x|, reduces to a form of Burgers equation by defining w(x) = ∫ xu(y)dy [10].0Thus, an initial blowup for the aggregation problem is the same as a a singularity in the slope for Burgersequation. Generically, Burgers singularities form by creating a |x| 1/3 power singularity in w, which gives a−2/3 power blowup in u. This corresponds to a blowup in L p for p > 3/2, but does not result in an initialmass concentration. However, as we well know, the Burgers solution forms a shock immediately afterwards,resulting in a delta concentration in u. Thus the scenario described above is a multidimensional analogueof the well-known behavior of how singularities initially form in Burgers equation. The delta-concentrationsare analogues of shock formation in scalar conservation laws.Section 4 constructs an example of a measure solution with initial data in L p , p < p s , that instantaneouslyconcentrates mass, for the special kernel K(x) = |x| (near the origin). We conjecture that such solutions existfor more general power-law kernels K(x) = |x| α , 2 − d < α < 2. The proof of instantaneous concentrationuses some monotonicity properties of the convolution operator ⃗x/|x| which would need to be proved for themore general case.25

Several interesting open problems remain, in addition to proving sharpness of the exponent p s for moregeneral kernels. For initial data in L ps local well-posedness is not known and we do not have intuition forthis.7 Appendix – Proof of Proposition 16Since all the convergences (2.38)-(2.40) take place on compact sets, one of the key ideas of this proof will beto approximate u 0 ∈ L p by a function with compact support and to use the fact that X t maps compact setsto compact sets (Lemma 14).PART I: We will prove that u(t, x) = u 0 (X −t (x))a(t, x) belongs to C([0, T ∗ ), L p (R d )). Assume first thatu 0 ∈ C c (R d ). The function u 0 is then uniformly continuous and, sinceit is clear that the quantitysup |X −t (x) − X −s (x)| ≤ C|t − s|,x∈R d‖u 0 (X −t (x)) − u 0 (X s (x))‖ L pcan be made as small as we want by choosing |t−s| small enough. We have therefore proven that u 0 (X −t (x)) ∈C([0, T ∗ ), L p (R d ). Assume now that u 0 ∈ L p (R d ). Approximate it by a function g ∈ C c (R d ) and write:‖u 0 (X −t (x)) − u 0 (X −s (x))‖ L p ≤ ‖u 0 (X −t (x)) − g(X −t (x))‖ L p + ‖g(X −t (x)) − g(X −s (x))‖ L p= I + II + III.+‖g(X −s (x)) − u 0 (X −s (x))‖ L pAs we have seen above, the second term can be made as small as we want by choosing |t − s| small enough.Using Lemma 13 we getI, III ≤ C ∗ ‖u 0 − g‖ L p,which can be made as small as we want since C c (R d ) is dense in L p (R d ). We have therefore proven that, ifu 0 ∈ L p (R d ), then(t, x) ↦→ u 0 (X −t (x)) ∈ C([0, T ∗ ), L p (R d )). (7.116)Let us now consider the function u 0 (X −t (x))a(t, x).[0, T ∗ ] × R d . Write‖u 0 (X −t (x))a(t, x) − u 0 (X −s (x))a(s, x)‖ L p ≤‖u 0 (X −t (x)){a(t, x) − a(s, x)}‖ L pRecall that a(t, x) is continuous and bounded on+‖{u 0 (X −t (x)) − u 0 (X −s (x))}a(s, x)‖ L p= I + II.Since a(s, x) is bounded, (7.116) implies that II can be made as small as we want by choosing |t − s| smallenough. Let us now take care of I. Approximate u 0 by a function g ∈ C c (R d ) and writeI≤ ‖{u 0 (X −t (x)) − g(X −t (x))}{a(t, x) − a(s, x)}‖ L p+‖g(X −t (x)){a(t, x) − a(s, x)}‖ L p= A + B.From Lemma 13 we haveA ≤ 2 C ∗ ‖u 0 − g‖ L psup |a(t, x)|,(t,x)∈[0,T ∗ ]×R d26

and since C c (R d ) is dense in L p , A can be made as small as we want. Let Ω denote the compact support ofg. Using Lemmas 13 and 14 we obtain:(∫B =(≤Ω+Ct∣∣g(X −t (x)){a(t, x) − a(s, x)} ∣ ) 1/pp dxsup |a(t, x) − a(s, x)|x∈Ω+ct)C ∗ ‖g‖ L p.Since a(t, x) is uniformly continuous on compact subset of [0, T ∗ ] × R d , it is clear that by choosing |t − s|small enough we can make B as small as we want. We have proven thatu ∈ C([0, T ∗ ], L p (R d ).PART II: In Part I we have proven that u ∈ C([0, T ∗ ], L p ). The same proof work to show that u ɛ , uvand u ɛ v ɛ belong to C([0, T ∗ ], L p ) (v(t, x) is continuous and bounded, so we can handle it exactly like a(t, x).)PART III: Let us now prove that u ɛ (t, x) = u ɛ 0(Xɛ−t (x))a ɛ (t, x) converges to u(t, x) = u 0 (X −t (x))a(t, x).For convenience we write ɛ instead of ɛ k . To do this, we will successively prove:u 0 (Xɛ −t (x)) → u 0 (X −t (x)) in C([0, T ∗ ), L p ), (7.117)u ɛ 0(Xɛ −t (x)) → u 0 (X −t (x)) in C([0, T ∗ ), L p ), (7.118)u ɛ 0(Xɛ −t (x))a ɛ (t, x) → u 0 (X −t (x))a(t, x) in C([0, T ∗ ), L p ). (7.119)To prove (7.117), approximate u 0 ∈ L p (R d ) by a function g ∈ C c (R d ) and write:sup t∈[0,T ∗ ) ‖u 0 (Xɛ −t (x) − u 0 (X −t (x))‖ L p ≤ sup t∈[0,T ∗ ) ‖u 0 (Xɛ−t (x)) − g(Xɛ −t (x))‖ L p +sup t∈[0,T ∗ ) ‖g(Xɛ−t (x)) − g(X −t (x))‖ L p + sup t∈[0,T ∗ ) ‖g(X −t (x)) − u 0 (X −t (x))‖ L p= I + II + III.From Lemma 13 it is clear that I and III can be made as small as we want by choosing an appropriatefunction g. Using Lemma 14, we see that, if Ω is the support of gII =sup ∣ g(X−tɛ (x)) − g(Xt∈[0,T(∫Ω+Ct−t (x)) ∣ 1/p dx) p .∗Using the uniform continuity of g together with the fact that Xɛ−t (x) converges uniformly to X −t (x) on[0, T ∗ ] × Ω + CT ∗ , we can make II as small as we want by choosing ɛ small enough.This concludes the proof of (7.117). To prove (7.118), writesup ‖u ɛ 0(Xɛ −t (x)) − u 0 (X −t (x))‖ L p ≤ sup ‖u ɛ 0(Xɛ−t (x)) − u 0 (Xɛ−t (x)‖ L pt∈[0,T ∗ )t∈[0,T ∗ )= I + II.+ sup ‖u 0 (Xɛ−t (x)) − u 0 (X −t (x))‖ L pt∈[0,T ∗ )From (7.117) we know that II can be made as small as we want by choosing ɛ small enough. Using Lemma13 we obtainI ≤ C ∗ ‖u ɛ 0 − u 0 ‖ L p → 0 as ɛ → 0.27

Let us now prove (7.119). Writesup ‖u ɛ 0(Xɛ−t (x))a ɛ (t, x) − u 0 (X −t (x))a(t, x)‖ L pt∈[0,T ∗ ]≤supt∈[0,T ∗ ]‖{u ɛ 0(Xɛ−t (x)) − u 0 (X −t (x))}a ɛ (t, x)‖ L p+ sup ‖u 0 (X −t (y)){a ɛ (t, x) − a(t, x)}‖ L pt∈[0,T ∗ ]= I + II.Since a ɛ (t, x) is uniformly bounded on [0, T ∗ ] × R d , it is clear from (7.118) that I → 0 as ɛ → 0. If u 0 is inC c (R d ), then it is easy to prove that II → 0 as ɛ → 0. If u 0 ∈ L p (R d ), then approximate it by g ∈ C c (R d )and proceed as before.Part IV: To prove that u ɛ kv ɛ kconverges to uv in C([0, T ∗ ], L p ), proceed exactly as in the proof of(7.119).AcknowledgmentsWe thank José Antonio Carrillo and Dejan Slepčev for useful comments. The authors were supportedby NSF grant DMS-0907931. JR acknowledge support from the project MTM2008-06349-C03-03 DGI-MCI (Spain) and 2009-SGR-345 from AGAUR-Generalitat de Catalunya. We thank the Centre de RecercaMatemàtica (Barcelona) for providing an excellent atmosphere for research.References[1] L.A. Ambrosio, N. Gigli, G. Savaré, Gradient flows in metric spaces and in the space of probabilitymeasures, Lectures in Mathematics, Birkhäuser, 2005.[2] J.D. Benamou, Y. Brenier, A computational Fluid Mechanics solution to the Monge-Kantorovichmass transfer problem, Numer. Math., 84 (2000), pp. 375–393.[3] D. Benedetto, E. Caglioti, M. Pulvirenti, A kinetic equation for granular media, RAIRO Modél.Math. Anal. Numér., 31, (1997), 615–641.[4] A. Bertozzi, J. Brandman, Finite-time blow-up of L ∞ -weak solutions of an aggregation equation, toappear in Comm. Math. Sci., special issue in honor of Andrew Majda’s 60th birthday.[5] A. Bertozzi, J.A. Carrillo, T. Laurent, Blowup in multidimensional aggregation equations withmildly singular interaction kernels, Nonlinearity, 22, (2009), pp. 683-710.[6] A. Bertozzi, T. Laurent, Finite-time blow-up of solutions of an aggregation equation in R n , Comm.Math. Phys., 274 (2007), pp. 717–735.[7] P. Biler and A. Woyczyński Global and Expoding Solutions for Nonlocal Quadratic Evolution Problems,SIAM J. Appl. Math., 59 (1998), pp. 845-869.[8] A. Blanchet, J. A. Carrillo, and N. Masmoudi, Infinite Time Aggregation for the Critical Patlak-Keller-Segel model in R 2 , to appear in Comm. Pure Appl. Math.[9] A. Blanchet, J. Dolbeault, and B. Perthame, Two-dimensional Keller-Segel model: optimalcritical mass and qualitative properties of the solutions, Electron. J. Differential Equations, (2006), No.44, 32 pp. (electronic).[10] M. Bodnar and J.J.L. Velázquez, An integro-differential equation arising as a limit of individualcell-based models, J. Differential Equations 222, (2006), pp. 341–380.28

[11] S. Boi, V. Capasso and D. Morale, Modeling the aggregative behavior of ants of the species Polyergusrufescens, Spatial heterogeneity in ecological models (Alcalá de Henares, 1998), Nonlinear Anal. RealWorld Appl., 1 (2000), pp. 163–176.[12] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. PureAppl. Math. 44, (1991), pp. 375–417.[13] M.P. Brenner, P. Constantin, L.P. Kadanoff, A. Schenkel and S.C. Venkataramani,Diffusion, attraction and collapse, Nonlinearity 12, (1999), pp. 1071–1098.[14] M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short rangeinteractions, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal,8 (2007), pp. 939–958.[15] M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlineardiffusion, Networks and Heterogenous Media, 3(4), (2008), pp. 749-785.[16] J.A. Carrillo, M. Difrancesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weakmeasure solutions, finite-time aggregation and confinement for nonlocal interaction equations, preprint.[17] J.A. Carrillo, R.J. McCann, and C. Villani, Kinetic equilibration rates for granular media andrelated equations: entropy dissipation and mass transportation estimates, Rev. Matemática Iberoamericana,19 (2003), pp. 1-48.[18] J.A. Carrillo, R.J. McCann, C. Villani, Contractions in the 2-Wasserstein length space andthermalization of granular media, Arch. Rat. Mech. Anal., 179 (2006), pp. 217–263.[19] J.A. Carrillo, J. Rosado Uniqueness of bounded solutions to Aggregation equations by optimaltransport methods, preprint.[20] J. Dolbeault and B. Perthame, Optimal critical mass in the two-dimensional Keller-Segel modelin R 2 , C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 611–616.[21] Qiang Du, Ping Zhang, Existence of weak solutions to some vortex density models, Siam J. Math.Anal., Vol. 34, No. 6, pp. 1279–1299.[22] W. Gangbo, R.J. McCann, The geometry of optimal transportation, Acta Math., 177 (1996),pp. 113161.[23] V. Gazi and K. Passino, Stability analysis of swarms, IEEE Trans. Auto. Control, 48 (2003), pp. 692-697.[24] D. Holm and V. Putkaradze, Formation of clumps and patches in self-aggregation of finite-sizeparticles , Physica D 220, (2006), pp. 183–196.[25] D. Holm and V. Putkaradze, Aggregation of finite size particles with variable mobility, Phys. Rev.Lett., (2005) 95:226106.[26] Y. Huang and A. L. Bertozzi, Self-similar blow-up solutions to an aggregation equation, preprint2009.[27] E.F. Keller and L.A. Segel, Initiation of slide mold aggregation viewed as an instability, J. Theor.Biol., 26 (1970).[28] T. Laurent, Local and Global Existence for an Aggregation Equation, Communications in PartialDifferential Equations, 32 (2007), pp. 1941-1964.29

[29] D. Li and J. Rodrigo, Finite-time singularities of an aggregation equation in R n with fractionaldissipation, Comm. Math. Phys., 287(2), (2009), pp. 687-703.[30] D. Li and J. Rodrigo, Refined blowup criteria and nonsymmetric blowup of an aggregation equation,Advances in Mathematics, 220(1), (2009), pp. 1717-1738.[31] H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Arch. Ration. Mech.Anal. 172, (2004), pp. 407–428.[32] D. Li and X. Zhang, On a nonlocal aggregation model with nonlinear diffusion, (2008) preprint.[33] F. Lin and P. Zhang, On the hydrodynamic limit of Ginzburg-Landau vortices, Discrete Contin.Dynam. Systems, 6 (2000), pp. 121–142.[34] G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math.Pures Appl. 86 (2006), pp. 68–79.[35] R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), pp. 153–179.[36] A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Bio. 38, (1999),pp. 534–570.[37] D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregationbehavior: from individuals to populations, J. Math. Biol., 50 (2005), pp. 49–66.[38] C.M. Topaz and A.L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biologicalgroups, SIAM J. Appl. Math., 65 (2004), pp. 152–174.[39] C.M. Topaz, A.L. Bertozzi, and M.A. Lewis, A nonlocal continuum model for biological aggregation,Bulletin of Mathematical Biology, 68(7), pp. 1601-1623, 2006.[40] G. Toscani, One-dimensional kinetic models of granular flows, RAIRO Modél. Math. Anal. Numér.,34, 6 (2000), pp. 1277–1291.[41] C. Villani, Topics in optimal transportation, volume 58 of Graduate Studies in Mathematics, AmericanMathematical Society, Providence, RI, 2003.[42] C. Villani, Optimal transport, old and new, Lecture Notes for the 2005 Saint-Flour summer school, toappear, Springer 2008.[43] M. Vishik, Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Ann.scient. Éc. Norm. Sup., 4e série, t. 32 (1999), pp. 769–812.[44] V. I. Yudovich, Non-stationary flow of an incompressible liquid., Zh. Vychisl. Mat. Mat. Fiz, 3 (1963),pp. 1032-1066.30