L p Theory for the Multidimensional Aggregation Equation

Transcription

1 L p Theory for the Multiimensional Aggregation Equation ANDREA L. BERTOZZI University of California - Los Angeles THOMAS LAURENT University of California - Los Angeles AND JESÚS ROSADO Universitat Autònoma e Barcelona Abstract We consier well-poseness of the aggregation equation t u + iv(uv) = 0, v = K u with initial ata in P 2 (R ) L p (R ), in imensions two an higher. We consier raially symmetric kernels where the singularity at the origin is of orer x α, α > 2, an prove local well-poseness in P 2 (R ) L p (R ) for sufficiently large p > p s. In the special case of K(x) = x, the exponent p s = /( 1) is sharp for local well-poseness, in that solutions can instantaneously concentrate mass for initial ata in P 2 (R ) L p (R ) with p < p s. We also give an Osgoo conition on the potential K(x) which guaranties global existence an uniqueness in P 2 (R ) L p (R ). c 2000 Wiley Perioicals, Inc. 1.1 Backgroun 1 Introuction The multiimensional aggregation equation (1.1) u + iv (uv) = 0, t (1.2) v = K u, (1.3) u(0) = u 0, arises in a number of moels for biological aggregation [11, 14, 15, 23, 27, 36, 35, 37, 38] as well as problems in materials science [25, 24] an granular meia [3, 17, 18, 31, 39]. The same equation with aitional iffusion has been consiere in [8, 9, 13, 20, 26, 28, 29, 30] although we o not consier that case in this paper. For the invisci case, much work has been one recently on the question of finite time blowup in equations of this type, from boune or smooth initial ata [10, 6, 4, 5]. A recent stuy [16] proves well-poseness of measure solutions for semi-convex kernels. Global existence (but not uniqueness) of measure solutions Communications on Pure an Applie Mathematics, Vol. 000, (2000) c 2000 Wiley Perioicals, Inc.

2 2 A. BERTOZZI, T. LAURENT, J. ROSADO has been proven in [32, 21] in two space imension when K is exactly the Newtonian Potential. Moreover, numerical simulations [7], of aggregations involving K(x) = x, exhibit finite time blowup from boune ata in which the initial singularity remains in L p for some p rather than forming a mass concentration at the initial blowup time. These facts together bring up the very interesting question of how these equations behave in general when we consier initial ata in L p, that may be locally unboune but oes not involve mass concentration. This work serves to provie a fairly complete theory of the problem in L p, although some interesting questions remain regaring critical p exponents for general kernels an for ata that lives precisely in L p s for the special kernel K(x) = x. The L p framework aopte in this paper allows us to make two significant avances in the unerstaning of the aggregation equation. First, it allows us to consier potentials which are more singular than the one which have been consiere up to now (with the exception of [32, 21], where they consier the Newtonian potential in 2D). In previous works, the potential K was often require to be at worst Lipschitz singular at the origin, i.e. K(x) x α with α 1 (see [27, 6, 5, 16]). In our L p framework it is possible to consier potentials whose singularity at the origin is of orer x α with α > 2. Such potentials might have a cusp (in 2D) or even blow up (in 3D) at the origin. Interestingly, in imension 3, x 2 is exactly the Newtonian potential. So we can rephrase our result by saying that we prove local existence an uniqueness when the singularity of the potential is better than that of the Newtonian potential. The secon important results proven in this paper concerns the specific an biologically relevant potential K(x) = x. For such a potential, a concept of measure solution is provie in [16]. In the present paper we ientify the critical regularity neee on the initial ata in orer to guaranty that the solution will stay absolutely continuous with respect to the Lebesgue measure at least for short time. To be more specific, we prove that solutions whose initial ata are in P 2 (R ) L p (R ) remain in P 2 (R ) L p (R ) at least for short time if p > /( 1). Here P 2 (R ) enotes probability measure with boune secon moment. On the other han for any p < /( 1) we are able to exhibit initial ata in P 2 (R ) L p (R ) for which a elta Dirac appears instantaneously in the solution the solution loses its absolute continuity with respect to the Lebesgue measure instantaneously. 1.2 Main results of the paper Below we state the main results of this paper an how they connect to previous results in the literature. Theorem 1.1 (well-poseness). Consier 1 < q < an p its Höler conjugate. Suppose K W 1,q (R ) an u 0 L p (R ) P 2 (R ) is nonnegative. Then there exists a time T > 0 an a unique nonnegative function u C([0,T ],L p (R )) C 1 ([0,T ],W 1,p (R ))

3 such that (1.4) (1.5) (1.6) L P THEORY FOR THE AGGREGATION EQUATION 3 u (t) + iv ( u(t)v(t) ) = 0 t [0,T ], v(t) = u(t) K t [0,T ], u(0) = u 0. Moreover the secon moment stays boune an the L 1 norm is conserve. Furthermore, if ess sup K < +, then we have global well-poseness. Theorem 1.1 is prove in sections 2 an 3. The fact that W 1,q 1 loc (R ) W 1,q 2 loc (R ) for q 1 q 2 allows us to make the following efinition: Definition 1.2 (critical exponents q s an p s ). Suppose K(x) is compactly supporte (or ecays exponentially fast as x ) an belongs to W 1,q (R ) for some q (1,+ ). Then there exists an exponent q s (1,+ ] such that K W 1,q (R ) for all q < q s an K / W 1,q (R ) for all q > q s. The Höler conjugate of this exponent q s is enote p s. The exponent q s quantifies the singularity of the potential. The more singular the potential, the smaller is q s. For potentials that behave like a power function at the origin, K(x) x α as x 0, the exponents are easily compute: (1.7) q s = 2 α, an p s =, if 2 < α < 2, (2 α) (1.8) q s = +, an p s = 1, if α 2. We obtain the following picture for power like potentials: Theorem 1.3 (Existence an uniqueness for power potential). Suppose K is compactly supporte (or ecays exponentially fast at infinity). Suppose also that K C 2 (R \{0}) an K(x) x α as x 0. (i) If 2 < α < 2 then the aggregation equation is locally well pose in P 2 (R ) L p (R ) for every p > p s. Moreover, it is not globally well pose in P 2 (R ) L p (R ). (ii) If α 2 then the aggregation equation is globally well pose in P 2 (R ) L p (R ) for every p > 1. As a consequence we have existence an uniqueness for all potentials which are less singular than the Newtonian potential K(x) = x 2 at the origin. In two imensions this inclues potentials with cusp such as K(x) = x 1/2. In three imensions this inclues potentials that blow up such as K(x) = x 1/2. From [5, 16] we know that the support of compactly supporte solutions shrinks to a point in finite time, proving the secon assertion in point (i) above. The first part of (i) an statement (ii) are irect corollary of Theorem 1.1, Definition 1.2 an the fact that α 2 implies K boune. In the case where α = 1, i.e. K(x) x as x 0, the previous Theorem gives local well poseness in P 2 (R ) L p (R ) for all p > p s = 1. The next Theorem

4 4 A. BERTOZZI, T. LAURENT, J. ROSADO shows that it is not possible to obtain local well poseness in P 2 (R ) L p (R ) for p < p s = 1. Theorem 1.4 (Critical p-exponent to generate instantaneous mass concentration). Suppose K(x) = x in a neighborhoo of the origin, an suppose K is compactly supporte (or ecays exponentially fast at infinity). Then, for any p < p s = 1, there exists initial ata in P 2 (R ) L p (R ) for which a elta Dirac appears instantaneously in the measure solution. In orer to make sense of the statement of the previous Theorem, we nee 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-poseness in P 2 (R ) of the aggregation equation with semi-convex potentials. The solutions in [16] are weak measure solutions - they are not necessarily absolutely continuous with respect to the Lebesgue measure. Theorems 1.3 an 1.4 give a sharp conition on the initial ata in orer for the solution to stay absolutely continuous with respect to the Lebesgue measure for short time. Theorem 1.4 is proven in Section 4. Finally, in section 5 we consier a class of potential that will be referre to as the class of natural potentials. A potential is sai to be natural if it satisfies that a) it is a raially symmetric potential, i.e.: K(x) = k( x ), b) it is smooth away from the origin an it s singularity at the origin is not worse than Lipschitz, c) it oesn t exhibit pathological oscillation at the origin, ) its erivatives ecay fast enough at infinity. All these conitions will be more rigorously state later. It will be shown that the graient of natural potentials automatically belongs to W 1,q for q <, therefore, using the results from the sections 2 an 3, we have local existence an uniqueness 1. in P 2 (R ) L p (R ), p > A natural potential is sai to be repulsive in the short range if it has a local maximum at the origin an it is sai 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 sai to be strictly repulsive (respectively strictly attractive) at the origin. The main theorem of section 5 is the following: Theorem 1.5 (Osgoo conition for global well poseness). Suppose K is a natural potential. (i) If K is repulsive in the short range, then the aggregation equation is globally well pose in P 2 (R ) L p (R ), p > /( 1). (ii) If K is strictly attractive in the short range, the aggregation equation is globally well pose in P 2 (R ) L p (R ), p > /( 1), if an only if (1.9) r 1 k is not integrable at 0. (r)

5 L P THEORY FOR THE AGGREGATION EQUATION 5 By globally well pose in P 2 (R ) L p (R ), we mean that for any initial ata in P 2 (R ) L p (R ) the unique solution of the aggregation equation will exist for all time an will stay in P 2 (R ) L p (R ) for all time. Notice that the exponent /( 1) is not sharp in this theorem. Conition (1.9) will be refere as the Osgoo conition. It is easy to unerstan why the Osgoo conition is relevant while stuying blowup: the quantity r T () = 0 k (r) can be thought as the amount of time it takes for a particle obeying the ODE Ẋ = K(X) to reach the origin if it starts at a istance from it. For a potential satisfying the Osgoo conition, T () = +, which means that the particle can not reach the origin in finite time. The Osgoo conition was alreay shown in [5] to be necessary an sufficient for global well poseness of L -solutions. Extension to L p requires L p estimates rather than L estimates. See also [41] for an example of the use of the Osgoo conition in the context of the Euler equations for incompressible flui. The only if part of statement (ii) was proven in [5] an [16]. In these two works it was shown that if (1.9) is not satisfie, then compactly supporte solutions will collapse into a point mass an therefore leave L p in finite time. In section 5 we prove statement (i) an the if part of statement (ii). 2 Existence of L p -solutions In this section we show that if the interaction potential satisfies (2.1) K W 1,q (R ), 1 < q < +, an if the initial ata is nonnegative an belongs to L p (R ) (p an q are Höler conjugates) then there exists a solution to the aggregation equation. Moreover, either this solution exists for all times, or its L p -norm blows up in finite time. The uality between L p an L q guarantees enough smoothness in the velocity fiel v = K u to efine characteristics. We use the characteristics to construct a solution. The argument is inspire by the existence of L solutions of the incompressible 2D Euler equations by Yuovich [42] an of L solutions of the aggregation equation [4]. Section 3 proves uniqueness provie u P 2. We prove in Theorem 2.13 of the present section that if u 0 P 2 then the solution stays in P 2. Finally we prove that if in aition to (2.1), we have (2.2) ess sup K < +, then the solution constructe exists for all time. Most of the section is evote to the proof of the following theorem:

6 6 A. BERTOZZI, T. LAURENT, J. ROSADO Theorem 2.1 (Local existence). Consier 1 < q < an p its Höler conjugate. Suppose K W 1,q (R ) an suppose u 0 L p (R ) is nonnegative. Then there exists a time T > 0 an a nonnegative function such that (2.3) (2.4) (2.5) u C([0,T ],L p (R )) C 1 ([0,T ],W 1,p (R )) u (t) + iv ( u(t)v(t) ) = 0 t [0,T ], v(t) = u(t) K t [0,T ], u(0) = u 0. Moreover the function t u(t) p Lp is ifferentiable an satisfies { (2.6) u(t) p } L = (p 1) t R p u(t,x) p iv v(t,x) x t [0,T ]. The choice of the space Y p := C([0,T ],L p (R )) C 1 ([0,T ],W 1,p (R )) is motivate by the fact that, if u Y p an K W 1,q, then the velocity fiel is automatically C 1 in space an time: Lemma 2.2. Consier 1 < q < an p its Höler conjugate. If K W 1,q (R ) an u Y p then (2.7) u K C 1 ( [0,T ] R ) an u K C 1 ([0,T ] R ) K W 1,q (R ) u Y p where the norm C 1 ([0,T ] R ) an Y p are efine by v C 1 ([0,T ] R ) = sup v + sup v (2.8) [0,T ] R [0,T ] R t + sup i=1 (2.9) u Yp = sup u(t) L p (R ) + sup u (t) W t [0,T ] t [0,T 1,p (R ). ] [0,T ] R v x i, Proof. Recall that the convolution between a L p -function an a L q -function is continuous an sup x R f g(x) f L p g L q Therefore, since K an K xi are in L q, the mapping f K f is a boune linear transformation from L p (R ) to C 1 (R ), where C 1 (R ) is enowe with the norm f C 1 = sup f (x) + x R i=1 sup f (x). x R x i Since 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) an w x i (t,x) are continuous on [0,T ] R. Let us

7 L P THEORY FOR THE AGGREGATION EQUATION 7 now show that w t (t,x) exists an is continuous on [0,T ] R. Since u (t) C([0,T ],W 1,p ) an K W 1,q, we have w t (t,x) = ( u (t) K ) (x) = u (t),τ x K where, enote the pairing between the two ual spaces W 1,p (R ) an W 1,q (R ), an τ x enote the translation by x. Since x τ x K is a continuous mapping from R to W 1,q it is clear that w t (t,x) is continuous with respect to space. The continuity with respect to time come from the continuity of u (t) with respect to time. Inequality (2.7) is easily obtaine. Remark 2.3. Let us point out that (2.3) inee makes sense, when unerstoo as an equality in W 1,p. Since v C([0,T ],C 1 (R )) one can easily check that uv C([0,T ],L p (R )). Also recall that the injection i : L p (R ) W 1,p (R ) an the ifferentiation xi : L p (R ) W 1,p (R ) are boune linear operators. Therefore it is clear that both u an iv(uv) belong to C([0,T ],W 1,p (R )). Equation (2.3) has to be unerstan as an equality in W 1,p. The rest of this section is organize as follows. First we give the basic a priori estimates in subsection 2.1. Then, in subsection 2.2, we consier a mollifie an cut-off version of the aggregation equation for which we have global existence of smooth an compactly supporte solutions. In subsection 2.3 we show that the characteristics of this approximate problem are uniformly Lipschitz continuous on [0,T ] R, where T > 0 is some finite time epening on u 0 L p. In subsection 2.4 we pass to the limit in C([0,T ],L p ). To o this we nee the uniform Lipschitz boun on the characteristics together with the fact that the translation by x, x τ x u 0, is a continuous mapping from R to L p (R ). In subsection 2.5 we prove three theorems. We first prove continuation of solutions. We then prove that L p - solutions which start in P 2 stay in P 2 as long as they exist. An finally we prove global existence in the case where K is boune from above. 2.1 A priori estimates Suppose u Cc 1 ((0,T ) R ) is a nonnegative function which satisfies (2.3)- (2.4) in the classical sense. Suppose also that K Cc (R ). Integrating by part, we obtain that for any p (1, + ): (2.10) u(t,x) p x = (p 1) u(t,x) p iv v(t,x)x t (0,T ). t R R As a consequence we have: (2.11) t u(t) p L p (p 1) iv v(t) L u(t) p Lp t (0,T ), an by Höler s inequality: (2.12) t u(t) p L p (p 1) K L q u(t) p+1 L p.

8 8 A. BERTOZZI, T. LAURENT, J. ROSADO We now erive L estimates for the velocity fiel v = K u an its erivatives. Höler s inequality easily gives (2.13) (2.14) v(t,x) u(t) L p K L q (t,x) (0,T ) R, v j (t,x) x i u(t) 2 K L p x i x j (t,x) (0,T ) R. L q Since v t = K u t = K iv(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 (2.13) gives (2.15) v t (t,x) u(t) 2 L p K L q K L q (t,x) (0,T ) R. 2.2 Approximate smooth compactly supporte solutions In this section we eal with a smooth version of equation (2.3)-(2.5). Suppose u 0 L p (R ), 1 < p < +, an K W 1,q (R ). Consier the approximate problem (2.16) (2.17) (2.18) u t + iv(uv) = 0 in (0,+ ) R, v = K ε u in (0,+ ) R, u(0) = u ε 0, where K ε = J ε K, u ε 0 = J εu 0 an J ε is an operator which mollifies an cuts-off, J ε f = ( f M Rε ) η ε where η ε (x) is a stanar mollifier: η ε (x) = 1 ( x ) ε η, η Cc (R ),η 0, η(x)x = 1, ε R an M Rε (x) is a stanar cut-off function: M Rε (x) = M( x R ε ), R ε as ε 0, M(x) = 1 if x 1, M Cc (R ), 0 < M(x) < 1 if 1 < x < 2, M = 0 if 2 x. Let τ x enote the translation by x, i.e.: τ x f (y) := f (y x). It is well known that given a fixe f L r (R ), 1 < r < +, the mapping x τ x f from R to L r (R ) is uniformly continuous. In (iv) of the next lemma we show a slightly stronger result which will be neee later. Lemma 2.4 (Properties of J ε ). Suppose f L r (R ), 1 < r < +, then (i) J ε f C c (R ), (ii) J ε f L r f L r, (iii) lim ε 0 J ε f f L r = 0,

9 L P THEORY FOR THE AGGREGATION EQUATION 9 (iv) The family of mappings x τ x J ε f from R to L r (R ) is equicontinuous, i.e.: for each δ > 0, there exists a η > 0 inepenent of ε such that if x y η, then τ x J ε f τ y J ε f L r δ. Proof. Statements (i) an (ii) are obvious. If f is compactly supporte, one can easily prove (iii) by noting that f M Rε = f for ε small enough. If f is not compactly supporte, (iii) is obtaine by approximating f by a compactly supporte function an by using (ii). Let us now turn to the proof of (iv). Using (ii) we obtain τ x J ε f J ε f L r (τ x M Rε )(τ x f ) M Rε f ) L r τ x M Rε M Rε L τ x f L r + M Rε L τ x f f L r. Because x τ x f is continuous, the secon term can be mae as small as we want by choosing x small enough. Since τ x M Rε M Rε L M Rε L x 1 R ε M L x, the first term can be mae as small as we want by choosing x small enough an inepenently of ε. Proposition 2.5 (Global existence of smooth compactly-supporte approximates). Given ε,t > 0, there exists a nonnegative function u C 1 c ((0,T ) R ) which satisfy (2.18) in the classical sense. Proof. Since u ε 0 an Kε belong to C c (R ), we can use theorem 3 p of [27] to get the existence of a function u ε satisfying (2.19) (2.20) (2.21) (2.22) u ε L (0,T ;H k ),ut ε L,(0,T ;H k 1 ) for all k, ut ε + iv (u ε ( K ε u ε )) = 0 in (0,T ) R, u ε (0) = u ε 0, u ε (t,x) 0 for a.e. (t,x) (0,T ), R. Statement (2.19) implies that u ε C((0,T );H k 1 ). Using the continuous embeing H k 1 (R ) C 1 (R ) for k large enough we fin that u ε an u ε x i, 1 i, are continuous on (0,T ) R. Finally, (2.20) shows that u ε t is also continuous on (0,T ) R. We have proven that u ε C 1 ((0,T ) R ). It is then obvious that v ε = K ε u ε C 1 ((0,T ) R ). Note moreover that v ε (x,t) u ε L (0,T ;L 2 ) K ε L 2 for all (t,x) (0,T ) R. This combine with the fact that v ε is in C 1 shows that the characteristics are well efine an propagate with finite spee. This proves that u ε is compactly supporte in (0,T ) R (because u ε 0 is compactly supporte in R ).

10 10 A. BERTOZZI, T. LAURENT, J. ROSADO 2.3 Stuy of the velocity fiel an the inuce flow map Note that K ε an u ε are in the right function spaces so that we can apply to them to the a priori estimates erive in section 2.1. In particular we have: t uε (t) L p (p 1) K L q u ε (t) p+1 L p, u ε (0) L p u 0 L p. Using Gronwall inequality an the estimate on the supremum norm of the erivatives erive in section 2.1 we obtain: Lemma 2.6 (uniform boun for the smooth approximates). There exists a time T > 0 an a constant C > 0, both inepenent of ε, such that (2.23) (2.24) u ε (t) L p C t [0,T ], v ε (t,x), v ε x i (t,x), v ε t (t,x) C (t,x) [0,T ] R. From (2.24) it is clear that the family {v ε } is uniformly Lipschitz on [0,T ] R, with Lipschitz constant C. We can therefore use the Arzela-Ascoli Theorem to obtain the existence of a continuous function v(t,x) such that (2.25) v ε v uniformly on compact subset of [0,T ] R. It is easy to check that this function v is also Lipschitz continuous with Lipschitz constant C. The Lipschitz an boune vector fiel v ε generates a flow map X ε (t,α), t [0,T ], α R : X ε (t,α) = v ε (X ε (t,α),t), t X ε (0,α) = α, where we enote by Xε t : R R the mapping α X ε (t,α) an by Xε t the inverse of Xε. t The uniform Lipschitz boun on the vector fiel implies uniform Lipschitz boun on the flow map an its inverse (see for example [4] for a proof of this statement) we therefore have: Lemma 2.7 (uniform Lipschitz boun on Xε t an Xε t ). There exists a constant C > 0 inepenent of ε such that: (I): for all t [0,T ] an for all x 1,x 2 R Xε(x t 1 ) Xε(x t 2 ) C x 1 x 2 an Xε t (x 1 ) Xε t (x 2 ) C x 1 x 2, (II): for all t 1,t 2 [0,T ] an for all x R X t 1 ε (x) X t 2 ε (x) C t 1 t 2 an X t 1 ε (x) X t 2 ε (x) C t 1 t 2.

11 L P THEORY FOR THE AGGREGATION EQUATION 11 The Arzela-Ascoli Theorem then implies that there exists mapping X t an X t such that X t ε k (x) X t (x) uniformly on compact subset of [0,T ] R, X t ε k (x) X t (x) uniformly on compact subset of [0,T ] R. Moreover it is easy to check that X t an X t inherit the Lipschitz bouns of X t ε an Xε t. Since the mapping X t : R R is Lipschitz continuous, by Raemecher s Theorem it is ifferentiable almost everywhere. Therefore it makes sense to consier its Jacobian matrix DX t (α). Because of Lemma 2.7-(i) we know that there exists a constant C inepenent of t an ε such that sup et DX t (α) C an sup et DXε(α) t C. αεr α R By the change of variable we then easily obtain the following Lemma: Lemma 2.8. The mappings f f X t an f f Xε t, t [0,T ], ε > 0, are boune linear operators from L p (R ) to L p (R ). Moreover there exists a constant C inepenent of t an ε such that f X t L p C f L p an f X t ε L p C f L p for all f L p (R ). Note that Lemma 2.7-(ii) implies an therefore This gives us the following lemma: X t ε(α) α Ct for all (t,α) [0,T ) R, X t (α) α Ct for all (t,α) [0,T ) R. Lemma 2.9. Let Ω be a compact subset of R, then X t ε(ω) Ω +Ct an X t (Ω) Ω +Ct, where the compact set Ω +Ct is efine by 2.4 Convergence in C([0,T ),L p ) Ω +Ct := {x R : ist (x,ω) Ct}. Since u ε an v ε = u ε K are C 1 functions which satisfy (2.26) u ε t + v ε u ε = (iv v ε )u ε an u ε (0) = u ε 0 we have the simple representation formula for u ε (t,x), t [0,T ), x R : u ε (t,x) = u ε 0(X t ε (x))e t 0 iv v ε (s,xε (t s) (x))s = u ε 0(X t (x)) a ε (t,x). Lemma There exists a function a(t,x) C 1 ([0,T ) R ) an a sequence ε k 0 such that (2.27) a ε k (t,x) a(t,x) uniformly on compact subset of [0,T ] R. ε

12 12 A. BERTOZZI, T. LAURENT, J. ROSADO Proof. By the Arzela-Ascoli Theorem, it is enough to show that the family b ε (t,x) := t 0 iv v ε (s,xε (t s) (x))s is equicontinuous an uniformly boune. The uniform bouneness simply come from the fact that iv v ε = u ε K ε u ε 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 inepenent of ε an t such that b ε (t,x 1 ) b ε (t,x 2 ) δ if x 1 x 2 η. First, note that by Höler s inequality we have b ε (t,x 1 ) b ε (t,x 2 ) t 0 u ε (s) L p τ ξ J ε K τ ζ J ε K L q s where ξ stans for Xε (t s) (x 1 ) an ζ for Xε (t s) (x 2 ). Then equicontinuity in space is a consequence of Lemma 2.4 (iv) together with the fact that Xε (t s) (x 1 ) Xε (t s) (x 2 ) C x 1 x 2 where C is inepenent of t,s an ε. Let us finally prove equicontinuity in time. First note that, assuming that t 1 < t 2, b ε (t 1,x) b ε (t 2,x) = t1 0 t2 t 1 iv v ε (s,x (t 1 s) (x)) iv v ε (s,x (t 2 s) (x))s iv v ε (s,x (t 2 s) ε (x))s. Since iv v ε is uniformly boune we clearly have t2 iv v ε (s,x (t 2 s) ε (x))s C t 1 t 2. t 1 The other term can be treate exactly as before, when we prove equicontinuity in space. Recall that the function u ε (t,x) = u ε 0(Xε t (x)) a ε (t,x) satisfies the ε-problem (2.18). We also have the following convergences: (2.28) (2.29) (2.30) (2.31) Define the function u ε 0 u 0 in L p (R ), X t ε k (x) X t (x) unif. on compact subset of [0,T ] R, a ε k (t,x) a(t,x) unif. on compact subset of [0,T ] R, v ε k (t,x) v(t,x) unif. on compact subset of [0,T ] R. (2.32) u(t,x) := u 0 (X t (x)) a(t,x).

13 L P THEORY FOR THE AGGREGATION EQUATION 13 Convergence (2.28)-(2.31) together with Lemma 2.8 an 2.9 allow us to prove the following proposition. Proposition : u,u ε,uv an u ε v ε all belong to the space C([0,T ),L p (R )). Moreover we have: (2.33) (2.34) u ε k u in C([0,T ),L p ), u ε k v ε k uv in C([0,T ),L p ). Proof. Straightforwar, see appenix at the en of the paper. We now turn to the proof of the main theorem of this section. Proof of Theorem 2.1. Let φ Cc (0,T ) be a scalar test function. It is obvious that u ε an v ε satisfy: (2.35) (2.36) (2.37) T 0 u ε (t)φ (t) t + T 0 iv ( u ε (t)v ε (t) ) φ(t) t = 0, v ε (t,x) = (u ε (t) K ε )(x) for all (t,x) [0,T ] R, u ε (0) = u ε 0, where the integrals in (2.35) are the integral of a continuous function from [0,T ] to the Banach space W 1,p (R ). Recall that the injection i : L p (R ) W 1,p (R ) an the ifferentiation xi : L p (R ) W 1,p (R ) are boune linear operators. Therefore (2.33) an (2.34) imply (2.38) u ε k u in C([0,T ],W 1,p (R )), iv [u ε k v ε k ] iv [uv] in C([0,T ],W 1,p (R )), which is more than enough to pass to the limit in relation (2.35). To pass to the limit in (2.36), it is enough to note that for all (t,x) [0,T ] R we have (2.39) (u ε (t) K ε ) (u(t) K)(x) u ε (t) u(t) L p K ε L q + u(t) L p K ε K L q, an finally it is trivial to pass to the limit in relation (2.37). Equation (2.35) means that the continuous function u(t) (continuous function with values in W 1,p (R )) satisfies (2.3) in the istributional sense. But (2.3) implies that the istributional erivative u (t) is itself a continuous function with value in W 1,p (R ). Therefore u(t) is ifferentiable in the classical sense, i.e, it belongs to C 1 ([0,T ],W 1,p (R )), an (2.3) is satisfie in the classical sense. We now turn to the proof of (2.6). The u ε s satisfy (2.10). Integrating over [0,t], t < T, we get t (2.40) u ε (t) p L p = uε 0 p L p (p 1) u ε (s,x) p iv v ε (s,x) xt. 0 R

14 14 A. BERTOZZI, T. LAURENT, J. ROSADO Proposition 2.11 together with the general inequality ( (2.41) f p g p L 1 2p f p 1 L p + g p 1 L ) p f g L p implies that (2.42) (2.43) (u ε ) p,u p C([0,T ], L 1 (R )), (u ε k ) p u p C([0,T ], L 1 (R )). On the other han, replacing K by K in (2.39) we see right away that (2.44) (2.45) Combining (2.42)-(2.45) we obtain (2.46) (2.47) ivv, ivv ε C([0,T ], L (R )), ivv ε k ivv in C([0,T ],L (R )). u p ivv, (u ε ) p ivv ε C([0,T ], L 1 (R )), (u ε k ) p ivv ε k u p ivv in C([0,T ],L 1 (R )). So we can pass to the limit in (2.40) to obtain t u(t) p L p = u 0 p L p (p 1) u(s,x) p iv v(s,x) xt. 0 R But (2.46) implies that the function t R u(t,x)p iv v(t,x) x is continuous, therefore the function t u(t) p Lp is ifferentiable an satisfies (2.6). 2.5 Continuation an conserve properties Theorem 2.12 (Continuation of solutions). The solution provie by Theorem 2.1 can be continue up to a time T max (0,+ ]. If T max < +, then lim sup t T max τ [0,t] u(τ) L p = + Proof. The proof is stanar. One just nees to use the continuity of the solution with respect to time. Theorem 2.13 (Conservation of mass/ secon moment). (i) Uner the assumption of Theorem 2.1, an if we assume moreover that u 0 L 1 (R ), then the solution u belongs to C([0,T ],L 1 (R )) an satisfies u(t) L 1 = u 0 L 1 for all t [0,T ]. (ii) Uner the assumption of Theorem 2.1, an if we assume moreover that u 0 has boune secon moment, then the secon moment of u(t) stays boune for all t [0,T ]. Proof. We just nee to revisit the proof of Proposition Since u 0 L 1 L p it is clear that (2.48) u ε 0 = J ε u 0 u 0 in L 1 (R ) L p (R ). Using convergences (2.48), (2.29), (2.30) an (2.31) we prove that (2.49) u ε k u in C([0,T ], L 1 L p ).

15 L P THEORY FOR THE AGGREGATION EQUATION 15 The proof is exactly the same than the one of Proposition Since the aggregation equation is a conservation law, it is obvious that the smooth approximates satisfy u ε (t) L 1 = u ε 0 L 1. Using (2.49) we obtain u(t) L 1 = u 0 L 1. We now turn to the proof of (ii). Since the smooth approximates u ε have compact support, their secon moment is clearly finite, an the following manipulation are justifie: x 2 u ε (t,x)x = 2 x v ε u ε (x) t R R ( ) 1/2 ( ) 1/2 2 x 2 u ε (t,x)x v ε 2 u ε (t,x)x R R ( ) 1/2 (2.50) C x 2 u ε (t,x)x. R Assume now that the secon moment of u 0 is boune. A simple computation shows that if η ε is raially symmetric, then x 2 η ε = x 2 +secon moment of η ε. Therefore x 2 u ε 0(x)x x 2 η ε (x) u 0 (x)x R R x 2 u 0 (x)x + x 2 η ε (x)x R R (2.51) x 2 u 0 (x)x + 1 for ε small enough. R Inequality (2.51) come from the fact that the secon moment of η ε goes to 0 as ε goes to 0. Estimate (2.50) together with (2.51) provie us with a uniform boun of the secon moment of the u ε (t) which only epens the secon moment of u 0. Since u ε converges to u in L 1, we obviously have, for a given R an t: x R x 2 u(t,x)x = lim ε 0 x R x 2 u ε (t,x)x limsup ε 0 R x 2 u ε (t,x)x. Since R is arbitrary, this show that the secon moment of u(t, ) is boune for all t for which the solution exists. Combining Theorem 2.12 an 2.13 together with equality (2.6) we get: Theorem 2.14 (Global existence when K is boune from above). Uner the assumption of Theorem 2.1, an if we assume moreover that u 0 L 1 (R ) an ess sup K < +, then the solution u exists for all times (i.e.: T max = + ). Proof. Equality (2.6) can be written { (2.52) u(t) p } L = (p 1) t R p u(t,x) p (u(t) K)(x) x. Since K is boune from above we have (2.53) (u(s) K)(x) (ess sup K) u(s,x)x = (ess sup K) u 0 L R 1.

16 16 A. BERTOZZI, T. LAURENT, J. ROSADO Combining (2.52), (2.53) an Gronwall inequality gives u(t) p L p u 0 p L p e(p 1)(ess sup K) u 0 L 1t, so the L p -norm can not blow-up in finite time which, because of Theorem 2.12, implies global existence. 3 Uniqueness of solutions in P 2 (R ) L p (R ) In this section we use an optimal transport argument to prove uniqueness of solutions in P 2 (R ) L p (R ), when K W 1,q for 1 < q < an p its Höler conjugate. To o that, we shall follow the steps in [19], where the authors exten the work of Loeper [33] to prove, among other, uniqueness of P 2 L -solutions of 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 constructe in previous sections are istribution solutions, i.e. they satisfy T ( ) ϕ (3.1) (t,x) + v(t,x) ϕ(t,x) u(t,x)xt = ϕ(0,x)u 0 (x)x t R N 0 R N for all ϕ C0 ([0,T ) R N )). A function u(t,x) satisfying (3.1) is sai to be a istribution solution to the continuity equation (2.3) with the given velocity fiel v(t,x) an initial ata u 0 (x). In fact, it is uniquely characterize by u(t,x)x = u 0 (x)x B X t (B) for all measurable set B R, see [1]. Here X t : R R is the flow map associate with the velocity fiel v(t,x) an X t is its inverse. In the optimal transport terminology this is equivalent to say that X t transports 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, 34, 22, 1] are put together. Theorem 3.1 ([19]). Let ρ 1 an ρ 2 be two probability measures on R N, such that they are absolutely continuous with respect to the Lebesgue measure an W 2 (ρ 1,ρ 2 ) <, an let ρ θ be an interpolation measure between ρ 1 an ρ 2, efine as in [33] by (3.2) ρ θ = ((θ 1)T + (2 θ)i R N) # ρ 1 for θ [1,2], where T is the optimal transport map between ρ 1 an ρ 2 ue to Brenier s theorem [12] an I R N is the ientity map. Then there exists a vector fiel ν θ L 2 (R N,ρ θ x) such that i. θ ρ θ + iv(ρ θ ν θ ) = 0 for all θ [1,2]. ii. ρ θ ν θ 2 x = W 2 R N 2 (ρ 1,ρ 2 ) for all θ [1,2].

17 L P THEORY FOR THE AGGREGATION EQUATION 17 iii. 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 Eucliean Wasserstein istance between two probability measures f,g P(R n ), { } 1/2 (3.3) W 2 ( f,g) = inf v x 2 Π(v,x), Π Γ R n R n where Π runs over the set of joint probability measures on R n R n with marginals f an g. Now we are reay to prove the uniqueness of solutions to the aggregation equation. Theorem 3.2 (Uniqueness). Let u 1, u 2 be two solutions of equation (2.3) in the interval [0,T ] with initial ata u 0 P 2 (R ) L p (R ), 1 < p <, such that their L p -norms remain boune for all t [0,T ]. Assume too that v is given by v = K u, with K such that K W 1,q (R N ), p an q conjugates. Then u 1 (t) = u 2 (t) for all 0 t T. Proof. Consier two characteristics flow maps, X 1 an X 2, such that u i = X i #u 0, i = 1, 2. Define the quantity (3.4) Q(t) := 1 X 1 (t) X 2 (t) 2 u 0 (x)x, 2 R N From [19, Remark 2.3], we have W2 2(u 1(t),u 2 (t)) 2Q(t) which we now prove is zero for all times, implying that u 1 = u 2. Now, to see that Q(t) 0 we compute the erivative of Q with respect to time. Q = X 1 X 2,v 1 (X 1 ) v 2 (X 2 ) u 0 (x)x t R N = X 1 X 2,v 1 (X 1 ) v 1 (X 2 ) u 0 (x)x R N + X 1 X 2,v 1 (X 2 ) v 2 (X 2 ) u 0 (x)x R N where the time variable has been omitte for clarity. The above argument is justifie because, ue to Lemma 2.2, the velocity fiel is C 1 an boune. Taking into account the Lipschitz properties of v into the first integral an using Höler inequality in the secon one, we can write (3.5) Q t CQ(t) + Q(t) 1 2 ( = CQ(t) + Q(t) 1 2 I(t) 1 2. R N v 1 (X 2 (t,x)) v 2 (X 2 (t,x)) 2 u 0 (x)x ) 1 2

18 18 A. BERTOZZI, T. LAURENT, J. ROSADO Now, in orer to estimate I(t), we use that the solutions are constructe transporting the initial ata through their flow maps, so we can write it as I(t) = K (u 1 u 2 )[X 2 (t,x)] 2 u 0 (x)x = R N K (u 1 u 2 )(x) 2 u 2 (x)x. R N Thus, taking an interpolation measure u θ between u 1 an u 2 an using Höler inequality an first statement of Theorem 3.1 we can get a boun for I(t) (3.6) ( ( 2 ) ) I(t) K 2q 1/q R N 1 θ u θ u 2 (t) L p 2 (3.7) 1 D 2 K (ν θ u θ ) 2 L θ u 2q 2 (t) L p, where ν θ L 2 (R N,u θ x) is a vector fiel, as escribe in Theorem 3.1. Let us work on the first term of the right han sie. Using Young inequality, for α such that q = 1/q + 1/α we obtain 2 2 (3.8) D 2 K (ν θ u θ ) 2 L θ D 1 2q 2 K 2 L q ν θ u θ 2 L α θ. 1 Note that q (1,+ ) implies α (1,2). Therefore we can use Höler inequality with conjugate exponents 2/(2 α) an 2/α to obtain (3.9) 1 ν θ u θ 2 L α = ( u θ α/2 u θ α/2 ν θ α ) 2/α ( ) (2 α)/α ( ) u θ α/(2 α) u θ ν θ 2 whence, since we can see from simple algebraic manipulations with the exponents α that 2 α = p, the conjugate of q, 2 2 ( ) (3.10) D 2 K (ν θ u θ ) 2 L θ D 2q 2 K 2 L q u θ L p u θ ν θ 2 θ. Therefore, using statements (ii) an (iii) of Theorem 3.1 we obtain (3.11) I(t) u 2 L p max{ u 1 L p, u 2 L p} D 2 K 2 L qw 2 2 (u 1,u 2 ) CQ(t). Finally, going back to (3.5) we see that Q t Q(t), whence, since Q(0) = 0, we can conclue Q(t) 0 an thus u 1 = u 2. The limiting case p = is the one stuie in [19]. Remark 3.3. Note that in orer to make the above argument rigorous, we nee the graient of the kernel to be at least C 1 when estimating I. It is not the case here, but we can still obtain the estimate using smooth approximations. let us efine I ε (t) = K ε (u 1 u 2 )[X 2 (t,x)] 2 u 0 (x)x R N 1

19 L P THEORY FOR THE AGGREGATION EQUATION 19 where K ε = J ε K (see section 2.2). Since K ε converges to K in L q, it is clear that K ε (u 1 u 2 ) converges pointwise to K (u 1 u 2 ). Using the ominate convergence theorem together with the fact that K ε (u 1 u 2 ) L is uniformly boune we get that I ε (t) converges to I(t) for every t (0,T ). On the other han, ue to the efinition of u θ we can write the ifference u 2 u 1 as the integral between 1 an 2 of θ u θ with respect to θ. Now, since the equation θ u θ + iv(u θ ν θ ) = 0 is satisfie in the sense of istribution, an K ε C c (R ), we can replace θ u θ for iv(ν θ u θ ) an pass the ivergence to the other term of the convolution, so that the equality 2 1 (D 2 K ε ν θ u θ )(x)θ = K ε (u 2 u 1 )(x) hols for all x R. The rest of the manipulations performe above are straightforwar with K ε. Passing to the limit in (3.11) is easy since D 2 K ε converges to D 2 K in L q. 4 Instantaneous mass concentration when K(x) = x In this section we consier the aggregation equation with an interaction potential equal to x in a neighborhoo of the origin an whose graient is compactly supporte (or ecay exponentially fast at infinity). The Laplacian of this kin of potentials has a 1/ x singularity at the origin, therefore K belongs to W 1,q (R ) if an only if q [1,). The Höler conjugate of is 1. Using the theory evelope in section 2 an 3 we therefore get local existence an uniqueness of solutions in P 2 (R ) L p (R ) for all p > 1. Here we stuy the case where the initial ata is in P 2 (R ) L p (R ) for p < 1. Given p < 1 we exhibit initial ata in P 2(R ) L p (R ) for which the solution instantaneously concentrates mass at the origin (i.e. a elta Dirac at the origin is create instantaneously). This shows that the existence theory evelope in section 2 an 3 is in some sense sharp. This also shows that it is possible for a solution to lose instantaneously its absolute continuity with respect to the Lebesgue measure. The solutions constructe in this section have compact support, hence we can simply consier K(x) = x without changing the behavior of the solution, given that if the solution has a small enough support, it only feels the part of the potential aroun the origin. We buil on the work evelope in [16] on global existence for measure solutions with boune secon moment: Theorem 4.1 (Existence an uniqueness of measure solutions [16]). Suppose that K(x) = x. Given µ 0 P 2 (R ), there exists a unique weakly continuous family of

20 20 A. BERTOZZI, T. LAURENT, J. ROSADO probability measures (µ t ) t (0,+ ) satisfying (4.1) (4.2) (4.3) t µ t + iv(µ t v t ) = 0 in D ((0, ) R ), v t = 0 K µ t, µ t converges weakly to µ 0 as t 0. Here 0 K is the unique element of minimal norm in the subifferential of K. Simply speaking, since K(x) = x is smooth away from the origin an raially symmetric, we have 0 K(x) = x x for x 0 an 0 K(0) = 0, an thus: (4.4) ( 0 K µ)(x) = y x x y x y µ(y). Note that, µ t being a measure, it is important for 0 K to be efine for every x R so that (4.2) makes sense. Equation (4.1) means that + ( ψ ) (4.5) 0 R t (x,t) + ψ(x,t) v t(x) µ t (x) t = 0, for all ψ C 0 (R (0,+ )). From (4.4) it is clear that v t (x) 1 for all x an t, therefore the above integral makes sense. The main Theorem of this section is the following: Theorem 4.2 (Instantaneous mass concentration). Consier the initial ata { L if x < 1, (4.6) u 0 (x) = x 1+ε 0 otherwise, where ε (0,1) an L := ( x <1 x ( 1+ε) x) 1 is a normalizing constant. Note that u 0 L p (R ) for all p [1, 1+ε ). Let (µ t) t (0,+ ) be the unique measure solution of the aggregation equation with interaction potential K(x) = x an with initial ata u 0. Then, for every t > 0 we have µ t ({0}) > 0, i.e., mass is concentrate at the origin instantaneously an the solution is no longer continuous with respect to the Lebesgue measure. Theorem 4.2 is a consequence of the following estimate on the velocity fiel: Proposition 4.3. Let (µ t ) t (0,+ ) be the unique measure solution of the aggregation equation with interaction potential K(x) = x an with initial ata (4.6). Then, for all t [0,+ ) the velocity fiel v t = 0 K µ t is focussing an there exists a constant C > 0 such that (4.7) v t (x) C x 1 ε for all t [0,+ ) an x B(0,1). By focussing, we mean that the velocity fiel points inwar, i.e. there exists a nonnegative function λ t : [0,+ ) [0,+ ) such that v t (x) = λ t ( x ) x x.

21 L P THEORY FOR THE AGGREGATION EQUATION Representation formula for raially symmetric measure solutions In this section, we show that for raially symmetric measure solutions, the characteristics are well efine. As a consequence, the solution to (2.3) can be expresse as the push forwar of the initial ata by the flow map associate with the ODE efining the characteristics. In the following the unit sphere {x R, x = 1} is enote by S an its surface area by ω. Definition 4.4. If µ P(R ) is a raially symmetric probability measure, then we efine ˆµ P([0, + )) by for all I B([0,+ )). ˆµ(I) = µ({x R : x I}) Remark 4.5. If a measure µ is raially symmetric, then µ({x}) = 0 for all x 0, an therefore K(x y)µ(y) = R \{x} K(x y)µ(y) R for all x 0. In other wors, for x 0, ( K µ)(x) is well efine espite the fact that K is not efine at x = 0. As a consequence ( 0 K µ)(x) = ( K µ)(x) if x 0 an ( 0 K µ)(0) = 0. Remark 4.6. If the raially symmetric measure µ is continuous with respect to the Lebesgue measure an has raially symmetric ensity u(x) = ũ( x ), then ˆµ is also continuous with respect to the Lebesgue measure an has ensity û, where (4.8) û(r) = ω r 1 ũ(r). Lemma 4.7 (Polar coorinate formula for the convolution). Suppose µ P(R ) is raially symmetric. Let K(x) = x, then for all x 0 we have: ( + ( ) ) x x (4.9) (µ K)(x) = φ ˆµ(ρ) ρ x where the function φ : [0,+ ) [ 1,1] is efine by (4.10) φ(r) = 1 re 1 y ω re 1 y e 1σ(y). 0 S Proof. This comes from simple algebraic manipulations. These manipulations are shown in [5]. In the next Lemma we state properties of the function φ efine in (4.10). Lemma 4.8 (Properties of the function φ). (i) φ is continuous an non-ecreasing on [0,+ ). Moreover φ(0) = 0, an lim r φ(r) = 1.

22 22 A. BERTOZZI, T. LAURENT, J. ROSADO (ii) φ(r) is O(r) as r 0. To be more precise: φ(r) (4.11) lim = 1 1 (y e 1 ) r 0 r ω S 2 σ(y). r>0 Proof. Consier the function F : [0,+ ) S [ 1,1] efine by (4.12) (r,y) re 1 y re 1 y e 1. Since F is boune, we have that φ(r) = 1 ω S F(r,y)σ(y) = 1 ω S \{e 1 } F(r, y)σ(y). If y S \{e 1 } then the function r F(r,y) is continuous on [0,+ ) an C on (0,+ ). An explicit computation shows then that (4.13) F 1 F(r,y)2 (r,y) = 0, r re 1 y thus φ is non-ecreasing an, by the Lebesgue ominate convergence, it is easy to see that φ is continuous, φ(0) = 0 an lim r φ(r) = 1, which prove (i). To prove (ii), note that the function F r (r,y) can be extene by continuity on [0,+ ). Therefore the right erivative with respect to r of F(r,y) is well efine: F(r,y) F(0,y) lim = 1 F(0,y)2 = 1 (y e 1 ) 2. r 0 r y r>0 an since F/ r is boune on (0,+ ) S \{e 1 }, we can now use the Lebesgue ominate convergence theorem to conclue: φ(r) lim r 0 r r>0 φ(r) φ(0) = lim r 0 r r>0 = lim r 0 r>0 1 ω F(r,y) F(0,y) S r σ(y) = 1 ω S 1 (y e 1 ) 2 σ(y). ( ) Remark 4.9. Note that for r > 0 the function ρ φ r ρ is non increasing an continuous. Inee, it is equal to 1 when ρ = 0 an it ecreases to 0 as ρ. In particular, the integral in (4.9) is well efine for any probability measure ˆµ P([0,+ )). Remark In imension two, it is easy to check that lim r 1 φ (r) = + which implies that the erivative of the function φ has a singularity at r = 1 an thus, that the function φ is not C 1.

23 L P THEORY FOR THE AGGREGATION EQUATION 23 Proposition 4.11 (Characteristic ODE). Let K(x) = x an let (µ t ) t [0,+ ) be a weakly continuous family of raially symmetric probability measures. Then the velocity fiel { ( K µ t )(x) if x 0 (4.14) v(x, t) = 0 if x = 0 is continuous on R \{0} [0 + ). Moreover, for every x R, there exists an absolutely continuous function t X t (x), t [0,+ ), which satisfies (4.15) (4.16) t X t(x) = v(x t (x),t) X 0 (x) = x. for a.e. t (0,+ ), Proof. From formula (4.9), Remark 4.9, an the weak continuity of the family (µ t ) t [0,+ ), we obtain continuity in time. The continuity in space simply comes from the continuity an bouneness of the function φ together with the Lebesgue ominate convergence theorem. Since v is continuous on R \{0} [0 + ) we know from the Peano theorem that given x R \{0}, the initial value problem (4.15)-(4.16) has a C 1 solution at least for short time. We want to see that it is efine for all time. For that, note that by a continuation argument, the interval given by Peano theorem can be extene as long as the solution stays in R \{0}. Then, if we enote by T x the maximum time so that the solution exists in [0,T x ) we have that either T x = an we are one, or T x < +, in which case clearly lim t Tx X t (x) = 0, an we can exten 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 constructe is continuous on [0,+ ), C 1 on [0,+ )\{T x } an satisfies (4.15) 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 (2.3) as a push-forwar of the initial ata. See [1] or [40] for a efinition of the push-forwar of a measure by a map. Proposition 4.12 (Representation formula). Let (µ t ) t [0,+ ) be a raially symmetric measure solution of the aggregation equation with interaction potential K(x) = x, an let X t : R R be efine by (4.14), (4.15) an (4.16). 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 ominate convergence theorem, that t X t #µ 0 is weakly continuous. Let us now prove that µ t := X t #µ 0 satisfies (4.5) for all ψ C 0 (R (0, )). Given that the test function

24 24 A. BERTOZZI, T. LAURENT, J. ROSADO ψ is compactly supporte, there exist T > 0 such that ψ(x, t) = 0 for all t T. We therefore have: (4.17) R ( ) 0 = ψ(x,t )µ T (x) ψ(x,0)µ 0 (x) = ψ(x T (x),t ) ψ(x,0) µ 0 (x). R R If we now take into account that from Proposition 4.11 the mapping t φ(x t (x),t) is absolutely continuous, we can rewrite (4.17) as T ( ) (4.18) 0 = t ψ(x t(x),t) t µ 0 (x) (4.19) (4.20) = = = R 0 T R T 0 T 0 0 ( ψ(x t (x),t) v(x t (x),t) + ψ R ( ψ(x t (x),t) v(x t (x),t) + ψ t (X t(x),t) t (X t(x),t) R ( ψ(x,t) v(x,t) + ψ ) t (x,t) µ t (x) t. ) t µ 0 (x) ) µ 0 (x) t The step from (4.19) to (4.20) hols because of the fact that v(x,t) 1, which justifies the use of the Fubini Theorem. Remark 4.13 (Representation formula in polar coorinates). Let µ t an X t be as in the previous proposition. Let R t : [0,+ ) [0,+ ) be the function such that X t (x) = R t ( x ). Then (4.21) ˆµ t = R t # ˆµ 0. Remark Since φ is nonnegative (Lemma 4.8), from (4.9), (4.14) an (4.15) we see that the function t X t (x) = R t ( x ) is non increasing. 4.2 Proof of Proposition 4.3 an Theorem 4.2 We are now reay to prove the estimate on the velocity fiel an the instantaneous concentration result. We start by giving a frozen in time estimate of the velocity fiel. Lemma Let K(x) = x, an let u 0 (x) be efine by (4.6) for some ε (0,1). Then there exist a constant C > 0 such that (4.22) (u 0 K)(x) C x 1 ε for all x B(0,1)\{0}. Proof. Note that if we o the change of variable s = x ρ that (4.23) (u 0 K)(x) = x + 0 φ(s) û 0 ( x s )s s 2. in equation (4.9), we fin

25 L P THEORY FOR THE AGGREGATION EQUATION 25 On the other han, using (4.8) we see that the û 0 (r) corresponing to the u 0 (x) efine by (4.6) is { ω (4.24) û 0 (r) = r if r < 1, ε 0 otherwise. Then, plugging (4.24) in (4.23) we obtain that for all x 0 (4.25) (u 0 K)(x) = ω x 1 ε + x φ(ρ) ρ. ρ2 ε In light of statement (ii) of Lemma 4.8, we see that the previous integral converges as x 0. Hence (u 0 K)(x) is O( x 1 ε ) as x 0 an (4.22) follows. Finally, the last piece we nee in orer to prove Proposition 4.3 from the previous lemma, is the following comparison principle: Lemma 4.16 (Temporal monotonicity of the velocity). Let (µ t ) t (0,+ ) be a raially symmetric measure solution of the aggregation equation with interaction potential K(x) = x. Then, for every x R \{0} the function is non ecreasing. t ( K µ t )(x) Proof. Combining (4.9) an (4.21) we see that ( + (4.26) (µ t K)(x) = φ( x ) R t (ρ) ) ˆµ 0(ρ). 0 Now, by Lemma 4.8, φ is non ecreasing an ue to Remark 4.13, t R t (ρ) is non increasing. Henceforth it is clear that (4.26) is itself non ecreasing. At this point, Proposition 4.3 follows as a simple consequence of the frozen in time estimate (4.22) together with Lemma 4.16, an we can give an easy proof for the main result we introuce at the beginning of the section. Proof of Theorem 4.2. Using the representation formula (Proposition 4.12) an the efinition of the push forwar we get µ t ({0}) = (X t #µ 0 )({0}) = µ 0 (Xt 1 ({0})). Then, note that the solution of the ODE ṙ = r 1 ε reaches zero in finite time. Therefore, from Proposition 4.3 an 4.11 we obtain that for all t > 0, there exists δ > 0 such that X t (x) = 0 for all x < δ. In other wors, for all t > 0, there exists δ > 0 such that B(0,δ) Xt 1 ({0}). Clearly, given our choice of initial conition, we have that µ 0 (B(0,δ)) > 0 if δ > 0, an therefore µ t ({0}) > 0 if t > 0.

26 26 A. BERTOZZI, T. LAURENT, J. ROSADO 5 Osgoo conition for global well-poseness This section consiers the global well-poseness of the aggregation equation in P 2 (R ) L p (R ), epening on the potential K. We start by giving a precise efinitions of natural potential, repulsive in the short range an strictly attractive in the short range, an then we prove Theorem 1.5. Definition 5.1. A natural potential is a raially symmetric potential K(x) = k( x ), where k : (0, + ) R is a smooth function which satisfies the following conitions: (C1) sup k (r) < +, r (0, ) (C2) α > such that k (r) an k (r) are O(1/r α ) as r +, (MN1) δ 1 > 0 such that k (r) is monotonic (either increasing or ecreasing) in (0,δ 1 ), (MN2) δ 2 > 0 such that rk (r) is monotonic (either increasing or ecreasing) in (0,δ 2 ). Remark 5.2. Note that monotonicity conition (MN1) implies that k (r) an k(r) are also monotonic in some (ifferent) neighborhoo of the origin (0, δ). Also, note that (C1) an (MN1) imply (C3) lim r 0 k (r) exists an is finite. + Remark 5.3. The far fiel conition (C2) can be roppe when the ata has compact support. Definition 5.4. A natural potential is sai to be repulsive in the short range if there exists an interval (0,δ) on which k(r) is ecreasing. A natural potential is sai to be strictly attractive in the short range if there exists an interval (0,δ) on which k(r) is strictly increasing. We woul like to remark that the two monotonicity conitions are not very restrictive as, in orer to violate them, a potential woul have to exhibit some pathological behavior aroun the origin, like oscillating faster an faster as r Properties of natural potentials As a last step before proving Theorem 1.5, let us point out some properties of natural potentials, which show the reason behin the choice of this kin of potentials to work with. Lemma 5.5. If K(x) = k( x ) is a natural potential, then k (r) = o(1/r) as r 0. Proof. First, note that since k(r) is smooth away from 0 we have k (1) k (ε) = 1 ε k (r)r.