L p Theory for the Multidimensional Aggregation Equation

Transcription

1 L p Theoy fo the Multiimensional Aggegation Equation Anea L. Betozzi, Thomas Lauent* & Jesus Rosao Novembe 19, 29 Abstact We consie well-poseness of the aggegation equation tu + iv(uv) =, v = K u with initial ata in P 2(R ) L p (R ), in imensions two an highe. We consie aially symmetic kenels whee the singulaity at the oigin is of oe x α, α > 2, an pove local well-poseness in P 2(R ) L p (R ) fo sufficiently lage p > p s. In the special case of K(x) = x, the exponent p s = /( 1) is shap fo local well-poseness, in that solutions can instantaneously concentate mass fo 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 guaanties global existence an uniqueness in P 2(R ) L p (R ). 1 Intouction 1.1 Backgoun The multiimensional aggegation equation u + iv (uv) =, t (1.1) v = K u, (1.2) u() = u, (1.3) aises in a numbe of moels fo biological aggegation [11, 14, 15, 23, 28, 37, 36, 38, 39] as well as poblems in mateials science [24, 25] an ganula meia [3, 17, 18, 31, 4]. The same equation with aitional iffusion has been consiee in [7, 9, 13, 2, 27, 29, 3, 32] although we o not consie that case in this pape. Fo the invisci case, much wok has been one ecently on the question of finite time blowup in equations of this type, fom boune o smooth initial ata [1, 6, 4, 5]. A ecent stuy [16] poves well-poseness of measue solutions fo semi-convex kenels. Global existence (but not uniqueness) of measue solutions has been poven in [33, 21] in two space imension when K is exactly the Newtonian Potential. Moeove, numeical simulations [26], of aggegations involving K(x) = x, exhibit finite time blowup fom boune ata in which the initial singulaity emains in L p fo some p athe than foming a mass concentation at the initial blowup time. These facts togethe bing up the vey inteesting question of how these equations behave in geneal when we consie initial ata in L p, that may be locally unboune but oes not involve mass concentation. This wok seves to povie a faily complete theoy of the poblem in L p, although some inteesting questions emain egaing citical p exponents fo geneal kenels an fo ata that lives pecisely in L ps fo the special kenel K(x) = x. The L p famewok aopte in this pape allows us to make two significant avances in the unestaning of the aggegation equation. Fist, it allows us to consie potentials which ae moe singula than the one which have been consiee up to now (with the exception of [33, 21], whee they consie the Newtonian Depatment of Mathematics, Univesity of Califonia - Los Angeles, Los Angeles, Califonia 995, USA betozzi@math.ucla.eu, Intenet: betozzi lauent@math.ucla.eu, Intenet: lauent Depatament e Matemàtiques, Univesitat Autònoma e Bacelona, E-8193 Bellatea, Spain. josao@mat.uab.cat, Intenet: josao 1

2 potential in 2D). In pevious woks, the potential K was often equie to be at wost Lipschitz singula at the oigin, i.e. K(x) x α with α 1 (see [28, 6, 5, 16]). In ou L p famewok it is possible to consie potentials whose singulaity at the oigin is of oe x α with α > 2. Such potentials might have a cusp (in 2D) o even blow up (in 3D) at the oigin. Inteestingly, in imension 3, x 2 is exactly the Newtonian potential. So we can ephase ou esult by saying that we pove local existence an uniqueness when the singulaity of the potential is bette than that of the Newtonian potential. The secon impotant esults poven in this pape concens the specific an biologically elevant potential K(x) = x. Fo such a potential, a concept of measue solution is povie in [16]. In the pesent pape we ientify the citical egulaity neee on the initial ata in oe to guaanty that the solution will stay absolutely continuous with espect to the Lebesgue measue at least fo shot time. To be moe specific, we pove that solutions whose initial ata ae in P 2 (R ) L p (R ) emain in P 2 (R ) L p (R ) at least fo shot time if p > /( 1). Hee P 2 (R ) enotes pobability measue with boune secon moment. On the othe han fo any p < /( 1) we ae able to exhibit initial ata in P 2 (R ) L p (R ) fo which a elta Diac appeas instantaneously in the solution the solution loses its absolute continuity with espect to the Lebesgue measue instantaneously. 1.2 Main esults of the pape Below we state the main esults of this pape an how they connect to pevious esults in the liteatue. Theoem 1 (well-poseness). Consie 1 < q < an p its Höle conjugate. Suppose K W 1,q (R ) an u L p (R ) P 2 (R ) is nonnegative. Then thee exists a time T > an a nonnegative function u C([, T ], L p (R )) C 1 ([, T ], W 1,p (R )) such that u (t) + iv ( u(t) v(t) ) = t [, T ], (1.4) v(t) = u(t) K t [, T ], (1.5) u() = u. (1.6) Moeove the secon moment stays boune an the L 1 nom is conseve. Futhemoe, if ess sup K < +, then we have global well-poseness. Theoem 1 is pove in sections 2 an 3. The fact that W 1,q1 loc (R ) W 1,q2 loc (R ) fo q 1 q 2 allows us to make the following efinition: Definition 2 (citical exponents q s an p s ). Suppose K(x) is compactly suppote (o ecays exponentially fast as x ) an belongs to W 1,q (R ) fo some q (1, + ). Then thee exists an exponent q s (1, + ] such that K W 1,q (R ) fo all q < q s an K / W 1,q (R ) fo all q > q s. The Höle conjugate of this exponent q s is enote p s. The exponent q s quantifies the singulaity of the potential. The moe singula the potential, the smalle is q s. Fo potentials that behave like a powe function at the oigin, K(x) x α as x, the exponents ae easily compute: q s = 2 α, an p s =, if 2 < α < 2, (1.7) (α 2) q s = +, an p s = 1, if α 2. (1.8) We obtain the following pictue fo powe like potentials: Theoem 3 (Existence an uniqueness fo powe potential). Suppose K is compactly suppote (o ecays exponentially fast at infinity). Suppose also that K C 2 (R \{}) an K(x) x α as x. (i) If 2 < α < 2 then the aggegation equation is locally well pose in P 2 (R ) L p (R ) fo evey p > p s. Moeove, it is not globally well pose in P 2 (R ) L p (R ). 2

3 (ii) If α 2 then the aggegation equation is globally well pose in P 2 (R ) L p (R ) fo evey p > 1. As a consequence we have existence an uniqueness fo all potentials which ae less singula than the Newtonian potential K(x) = x 2 at the oigin. In two imensions this inclues potentials with cusp such as K(x) = x 1/2. In thee imensions this inclues potentials that blow up such as K(x) = x 1/2. Fom [5, 16] we know that the suppot of compactly suppote solutions shinks to a point in finite time, poving the secon assetion in point (i) above. The fist pat of (i) an statement (ii) ae iect coollay of Theoem 1, Definition 2 an the fact that α 2 implies K boune. In the case whee α = 1, i.e. K(x) x as x, the pevious Theoem gives local well poseness in P 2 (R ) L p (R ) fo all p > p s = 1. The next Theoem shows that it is not possible to obtain local well poseness in P 2 (R ) L p (R ) fo p < p s = 1. Theoem 4 (Citical p-exponent to geneate instantaneous mass concentation). Suppose K(x) = x in a neighbohoo of the oigin, an suppose K is compactly suppote (o ecays exponentially fast at infinity). Then, fo any p < p s = 1, thee exists initial ata in P 2(R ) L p (R ) fo which a elta Diac appeas instantaneously in the measue solution. In oe to make sense of the statement of the pevious Theoem, we nee a concept of measue solution. The potentials K(x) = x is semi-convex, i.e. thee exist λ R such that K(x) λ 2 x 2 is convex. In [16], Caillo et al. pove global well-poseness in P 2 (R ) of the aggegation equation with semi-convex potentials. The solutions in [16] ae weak measue solutions - they ae not necessaily absolutely continuous with espect to the Lebesgue measue. Theoems 3 an 4 give a shap conition on the initial ata in oe fo the solution to stay absolutely continuous with espect to the Lebesgue measue fo shot time. Theoem 4 is poven in Section 4. Finally, in section 5 we consie a class of potential that will be efee to as the class of natual potentials. A potential is sai to be natual if it satisfies that a) it is a aially symmetic potential, i.e.: K(x) = k( x ), b) it is smooth away fom the oigin an it s singulaity at the oigin is not wose than Lipschitz, c) it oesn t exhibit pathological oscillation at the oigin, ) its eivatives ecay fast enough at infinity. All these conitions will be moe igoously state late. It will be shown that the gaient of natual potentials automatically belongs to W 1,q fo q <, theefoe, using the esults fom the sections 2 an 3, we have local existence an uniqueness in P 2 (R ) L p (R ), p > 1. A natual potential is sai to be epulsive in the shot ange if it has a local maximum at the oigin an it is sai to be attactive in the shot ange if it has a local minimum at the oigin. If the maximum (espectively minimum) is stict, the natual potential is sai to be stictly epulsive (espectively stictly attactive) at the oigin. The main theoem of section 5 is the following: Theoem 5 (Osgoo conition fo global well poseness). Suppose K is a natual potential. (i) If K is epulsive in the shot ange, then the aggegation equation is globally well pose in P 2 (R ) L p (R ), p > /( 1). (ii) If K is stictly attactive in the shot ange, the aggegation equation is globally well pose in P 2 (R ) L p (R ), p > /( 1), if an only if 1 k () is not integable at. (1.9) 3

4 By globally well pose in P 2 (R ) L p (R ), we mean that fo any initial ata in P 2 (R ) L p (R ) the unique solution of the aggegation equation will exist fo all time an will stay in P 2 (R ) L p (R ) fo all time.notice that the exponent /( 1) is not shap in this theoem. Conition (1.9) will be efee as the Osgoo conition. It is easy to unestan why the Osgoo conition is elevant while stuying blowup: the quantity T () = k () can be thought as the amount of time it takes fo a paticle obeying the ODE Ẋ = K(X) to each the oigin if it stats at a istance fom it. Fo a potential satisfying the Osgoo conition, T () = +, which means that the paticle can not each the oigin in finite time. The Osgoo conition was aleay shown in [5] to be necessay an sufficient fo global well poseness of L -solutions. Extension to L p equies L p estimates athe than L estimates. See also [43] fo an example of the use of the Osgoo conition in the context of the Eule equations fo incompessible flui. The only if pat of statement (ii) was poven in [5] an [16]. In these two woks it was shown that if (1.9) is not satisfie, then compactly suppote solutions will collapse into a point mass an theefoe leave L p in finite time. In section 5 we pove statement (i) an the if pat of statement (ii). 2 Existence of L p -solutions In this section we show that if the inteaction potential satisfies K W 1,q (R ), 1 < q < +, (2.1) an if the initial ata is nonnegative an belongs to L p (R ) (p an q ae Höle conjugates) then thee exists a solution to the aggegation equation. Moeove, eithe this solution exists fo all times, o its L p -nom blows up in finite time. The uality between L p an L q guaantees enough smoothness in the velocity fiel v = K u to efine chaacteistics. We use the chaacteistics to constuct a solution. The agument is inspie by the existence of L solutions of the incompessible 2D Eule equations by Yuovich [44] an of L solutions of the aggegation equation [4]. Section 3 poves uniqueness povie u P 2. We pove in Theoem 18 of the pesent section that if u P 2 then the solution stays in P 2. Finally we pove that if in aition to (2.1), we have ess sup K < +, (2.11) then the solution constucte exists fo all time. Most of the section is evote to the poof of the following theoem: Theoem 6 (Local existence). Consie 1 < q < an p its Höle conjugate. Suppose K W 1,q (R ) an suppose u L p (R ) is nonnegative. Then thee exists a time T > an a nonnegative function u C([, T ], L p (R )) C 1 ([, T ], W 1,p (R )) such that u (t) + iv ( u(t) v(t) ) = t [, T ], (2.12) v(t) = u(t) K t [, T ], (2.13) u() = u. (2.14) Moeove the function t u(t) p L is iffeentiable an satisfies p t { u(t) p Lp} = (p 1) u(t, x) p iv v(t, x) x t [, T ]. (2.15) R The choice of the space Y p := C([, T ], L p (R )) C 1 ([, 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: 4

5 Lemma 7. Consie 1 < q < an p its Höle conjugate. If K W 1,q (R ) an u Y p then u K C 1 ( [, T ] R ) an u K C1 ([,T ] R ) K W 1,q (R ) u Y p (2.16) whee the nom C 1 ([,T ] R ) an Y p ae efine by v C1 ([,T ] R ) = sup v + [,T ] R sup [,T ] R u Yp = sup u(t) Lp (R ) + t [,T ] sup t [,T ] v t + sup i=1 [,T ] R v x i, (2.17) u (t) W 1,p (R ). (2.18) Poof. 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 Theefoe, since K an K xi ae in L q, the mapping f K f is a boune linea tansfomation fom L p (R ) to C 1 (R ), whee C 1 (R ) is enowe with the nom f C 1 = sup x R f(x) + sup f (x). x R x i Since u C([, T ], L p ) it is then clea that u K C([, T ], C 1 ). In paticula w(t, x) = (u(t) K) (x) an w x i (t, x) ae continuous on [, T ] R. Let us now show that w t (t, x) exists an is continuous on [, T ] R. Since u (t) C([, T ], W 1,p ) an K W 1,q, we have i=1 w t (t, x) = (u (t) K) (x) = u (t), τ x K whee, enote the paiing between the two ual spaces W 1,p (R ) an W 1,q (R ), an τ x enote the tanslation by x. Since x τ x K is a continuous mapping fom R to W 1,q it is clea that w t (t, x) is continuous with espect to space. The continuity with espect to time come fom the continuity of u (t) with espect to time. Inequality (2.16) is easily obtaine. Remak 8. Let us point out that (2.12) inee makes sense, when unestoo as an equality in W 1,p. Since v C([, T ], C 1 (R )) one can easily check that uv C([, T ], L p (R )). Also ecall that the injection i : L p (R ) W 1,p (R ) an the iffeentiation xi : L p (R ) W 1,p (R ) ae boune linea opeatos. Theefoe it is clea that both u an iv(uv) belong to C([, T ], W 1,p (R )). Equation (2.12) has to be unestan as an equality in W 1,p. The est of this section is oganize as follows. Fist we give the basic a pioi estimates in subsection 2.1. Then, in subsection 2.2, we consie a mollifie an cutte-off vesion of the aggegation equation fo which we have global existence of smooth an compactly suppote solutions. In subsection 2.3 we show that the chaacteistics of this appoximate poblem ae unifomly Lipschitz continuous on [, T ] R, whee T > is some finite time epening on u L p. In subsection 2.4 we pass to the limit in C([, T ], L p ). To o this we nee the unifom Lipschitz boun on the chaacteistics togethe with the fact that the tanslation by x, x τ x u, is a continuous mapping fom R to L p (R ). In subsection 2.5 we pove thee theoems. We fist pove continuation of solutions. We then pove that L p -solutions which stat in P 2 stay in P 2 as long as they exist. An finally we pove global existence in the case whee K is boune fom above. 2.1 A pioi estimates Suppose u Cc 1 ((, T ) R ) is a nonnegative function which satisfies (2.12)-(2.13) in the classical sense. Suppose also that K Cc (R ). Integating by pat, we obtain that fo any p (1, + ): u(t, x) p x = (p 1) u(t, x) p iv v(t, x)x t (, T ). (2.19) t R R 5

6 As a consequence we have: t u(t) p L (p 1) iv v(t) p L u(t) p Lp t (, T ), (2.2) an by Höle s inequality: t u(t) p L (p 1) K p L q u(t) p+1 L. (2.21) p We now eive L estimates fo the velocity fiel v = K u an its eivatives. Höle s inequality easily gives Since v t u = K t v(t, x) u(t) L p K L q (t, x) (, T ) R, (2.22) v j (t, x) x i u(t) L p 2 K x i x j (t, x) (, T ) R. (2.23) L q = K iv(uv) = K uv we have v (t, x) t 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, x) t u(t) 2 L p K L q K L q (t, x) (, T ) R. (2.24) 2.2 Appoximate smooth compactly suppote solutions In this section we eal with a smooth vesion of equation (2.12)-(2.14). Suppose u L p (R ), 1 < p < +, an K W 1,q (R ). Consie the appoximate poblem u t + iv(uv) = in (, + ) R, (2.25) v = K ɛ u in (, + ) R, (2.26) u() = u ɛ, (2.27) whee K ɛ = J ɛ K, u ɛ = J ɛ u an J ɛ is an opeato which mollifies an cuts-off, J ɛ f = (fm Rɛ ) η ɛ whee η ɛ (x) is a stana mollifie: η ɛ (x) = 1 ( x ) ɛ η, η Cc (R ), η, η(x)x = 1, ɛ R an M Rɛ (x) is a stana cut-off function: M Rɛ (x) = M( x R ɛ ), R ɛ as ɛ, M C c (R ), Let τ x enote the tanslation by x, i.e.: M(x) = 1 if x 1, < M(x) < 1 if 1 < x < 2, M = if 2 x. τ x f(y) := f(y x). It is well known that given a fixe f L (R ), 1 < < +, the mapping x τ x f fom R to L (R ) is unifomly continuous. In (iv) of the next lemma we show a slightly stonge esult which will be neee late. Lemma 9 (Popeties of J ɛ ). Suppose f L (R ), 1 < < +, then 6

7 (i) J ɛ f C c (R ), (ii) J ɛ f L f L, (iii) lim ɛ J ɛ f f L =, (iv) The family of mappings x τ x J ɛ f fom R to L (R ) is equicontinuous, i.e.: fo each δ >, thee is a η > inepenent of ɛ such that τ x J ɛ f τ y J ɛ f L δ if x y η. Poof. Statements (i) an (ii) ae obvious. If f is compactly suppote, one can easily pove (iii) by noting that fm Rɛ = f fo ɛ small enough. If f is not compactly suppote, (iii) is obtaine by appoximating f by a compactly suppote function an by using (ii). Let us now tun to the poof of (iv). Using (ii) we obtain τ x J ɛ f J ɛ f L (τ x M Rɛ )(τ x f) M Rɛ f) L τ x M Rɛ M Rɛ L τ x f L + M Rɛ L τ x f f L. Because x τ x f is continuous, the secon tem can be mae as small as we want by choosing x small enough. Since τ x M Rɛ M Rɛ L M Rɛ L x 1 R ɛ M L x, the fist tem can be mae as small as we want by choosing x small enough an inepenently of ɛ. Poposition 1 (Global existence of smooth compactly-suppote appoximates). Given ɛ, T >, thee exists a nonnegative function u C 1 c ((, T ) R ) which satisfy (2.27) in the classical sense. Poof. Since u ɛ an K ɛ belong to Cc function u ɛ satisfying (R ), we can use theoem 3 p of [28] to get the existence of a u ɛ L (, T ; H k ), u ɛ t L, (, T ; H k 1 ) fo all k, (2.28) u ɛ t + iv (u ɛ ( K ɛ u ɛ )) = in (, T ) R, (2.29) u ɛ () = u ɛ, (2.3) u ɛ (t, x) fo a.e. (t, x) (, T ), R. (2.31) Statement (2.28) implies that u ɛ C((, T ); H k 1 ). Using the continuous embeing H k 1 (R ) C 1 (R ) fo k lage enough we fin that u ɛ an u ɛ x i, 1 i, ae continuous on (, T ) R. Finally, (2.29) shows that u ɛ t is also continuous on (, T ) R. We have poven that u ɛ C 1 ((, T ) R ). It is then obvious that v ɛ = K ɛ u ɛ C 1 ((, T ) R ). Note moeove that v ɛ (x, t) u ɛ L (,T ;L 2 ) K ɛ L 2 fo all (t, x) (, T ) R. This combine with the fact that v ɛ is in C 1 shows that the chaacteistics ae well efine an popagate with finite spee. This poves that u ɛ is compactly suppote in (, T ) R (because u ɛ is compactly suppote in R ). 2.3 Stuy of the velocity fiel an the inuce flow map Note that K ɛ an u ɛ ae in the ight function spaces so that we can apply to them to the a pioi estimates eive in section 2.1. In paticula we have: t uɛ (t) L p (p 1) K L q u ɛ (t) p+1 L p, u ɛ () L p u L p. Using Gonwall inequality an the estimate on the supemum nom of the eivatives eive in section 2.1 we obtain: 7

8 Lemma 11 (unifom boun fo the smooth appoximates). Thee exists a time T > an a constant C >, both inepenent of ɛ, such that u ɛ (t) L p C t [, T ], (2.32) v ɛ t, x), v ɛ x i (t, x), v ɛ t(t, x) C (t, x) [, T ] R. (2.33) Fom (2.33) it is clea that the family {v ɛ } is unifomly Lipschitz on [, T ] R, with Lipschitz constant C. We can theefoe use the Azela-Ascoli Theoem to obtain the existence of a continuous function v(t, x) such that v ɛ v unifomly on compact subset of [, T ] R. (2.34) It is easy to check that this function v is also Lipschitz continuous with Lipschitz constant C. The Lipschitz an boune vecto fiel v ɛ geneates a flow map X ɛ (t, α), t [, T ], α R : X ɛ (t, α) = v ɛ (X ɛ (t, α), t), t X ɛ (, α) = α, whee we enote by X t ɛ : R R the mapping α X ɛ (t, α) an by X t ɛ the invese of X t ɛ. The unifom Lipschitz boun on the vecto fiel implies unifom Lipschitz boun on the flow map an its invese (see fo example [4] fo a poof of this statement) we theefoe have: Lemma 12 (unifom Lipschitz boun on Xɛ t an Xɛ t ). Thee exists a constant C > inepenent of ɛ such that: (i) fo all t [, T ] an fo all x 1, x 2 R Xɛ(x t 1 ) Xɛ(x t 2 ) C x 1 x 2 an Xɛ t (x 1 ) x t ɛ (x 2 ) C x 1 x 2, (ii) fo all t 1, t 2 [, T ] an fo all x R X t1 ɛ (x) X t2 ɛ (x) C t 1, t 2 an Xɛ t1 (x) X t2 ɛ (x) C t 1 t 2. The Azela-Ascoli Theoem then implies that thee exists mapping X t an X t such that X t ɛ k (x) X t (x) unifomly on compact subset of [, T ] R, X t ɛ k (x) X t (x) unifomly on compact subset of [, T ] R. Moeove it is easy to check that X t an X t inheit the Lipschitz bouns of X t ɛ an X t ɛ. Since the mapping X t : R R is Lipschitz continuous, by Raemeche s Theoem it is iffeentiable almost eveywhee. Theefoe it makes sense to consie its Jacobian matix DX t (α). Because of Lemma 12-(i) we know that thee exists a constant C inepenent of t an ε such that sup αɛr et DX t (α) C an sup α R et DX t ɛ(α) C. By the change of vaiable we then easily obtain the following Lemma: Lemma 13. The mappings f f X t an f f Xɛ t, t [, T ], ɛ >, ae boune linea opeatos fom L p (R ) to L p (R ). Moeove thee exists a constant C inepenent of t an ɛ such that f X t L p C f L p an f X t ɛ L p C f L p fo all f L p (R ). 8

9 Note that Lemma 12-(ii) implies an theefoe This gives us the following lemma: Lemma 14. let Ω be a compact subset of R, then whee the compact set Ω + Ct is efine by 2.4 Convegence in C([, T ), L p ) X t ɛ(α) α Ct fo all (t, α) [, T ) R, X t (α) α Ct fo all (t, α) [, T ) R. X t ɛ(ω) Ω + Ct an X t (Ω) Ω + Ct, Ω + Ct := {x R : ist (x, Ω) Ct}. Since u ɛ an v ɛ = u ɛ K ae C 1 functions which satisfy u ɛ t + v ɛ u ɛ = (iv v ɛ )u ɛ an u ɛ () = u ɛ (2.35) we have the simple epesentation fomula fo u ɛ (t, x), t [, T ), x R : u ɛ (t, x) = u ɛ (Xɛ t (x))e R t iv vɛ (s,x (t s) ɛ (x))s = u ɛ (X t (x)) a ɛ (t, x). Lemma 15. Thee exists a function a(t, x) C 1 ([, T ) R ) an a sequence ɛ k such that a ɛ k (t, x) a(t, x) unifomly on compact subset of [, T ] R. (2.36) Poof. By the Azela-Ascoli Theoem, it is enough to show that the family b ɛ (t, x) := t iv v ɛ (s, Xɛ (t s) (x))x is equicontinuous an unifomly boune. The unifom bouneness simply come fom the fact that iv v ɛ = u ɛ K ɛ u ɛ L p K L q. Let us now pove equicontinuity in space, i.e., we want to pove that fo each δ >, thee is η > inepenent of ɛ an t such that b ɛ (t, x 1 ) b ɛ (t, x 2 ) δ if x 1 x 2 η. Fist, note that by Höle s inequality we have ɛ b ɛ (t, x 1 ) b ɛ (t, x 2 ) t u ɛ (s) L p τ ξ J ɛ K τ ζ J ɛ K L q s whee ξ stans fo Xɛ (t s) (x 1 ) an ζ fo Xɛ (t s) (x 2 ). Then equicontinuity in space is a consequence of Lemma 9 (iv) togethe with the fact that whee C is inepenent of t, s an ɛ. Xɛ (t s) (x 1 ) Xɛ (t s) (x 2 ) C x 1 x 2 9

10 Let us finally pove equicontinuity in time. Fist not that, assuming that t 1 < t 2, b ɛ (t 1, x) b ɛ (t 2, x) = t1 t2 t 1 iv v ɛ (s, X (t1 s) (x)) iv v ɛ (s, X (t2 s) (x))s iv v ɛ (s, Xɛ (t2 s) (x))s. Since iv v ɛ is unifomly boune we clealy have t2 iv v ɛ (s, Xɛ (t2 s) (x))s C t 1 t 2. t 1 The othe tem can be teate exactly as befoe, when we pove equicontinuity in space. Recall that the function u ɛ (t, x) = u ɛ (Xɛ t (x)) a ɛ (t, x) satisfies the ɛ-poblem (2.27). We also have the following convegences: Define the function u ɛ u in L p (R ), (2.37) Xɛ t k (x) X t (x) unif. on compact subset of [, T ] R, (2.38) a ɛ k (t, x) a(t, x) unif. on compact subset of [, T ] R, (2.39) v ɛ k (t, x) v(t, x) unif. on compact subset of [, T ] R. (2.4) u(t, x) := u (X t (x)) a(t, x). (2.41) Convegence (2.37)-(2.4) togethe with Lemma 13 an 14 allow us to pove the following poposition. Poposition 16. : u, u ɛ, uv an u ɛ v ɛ all belong to the space C([, T ), L p (R )). Moeove we have: Poof. Staight fowa, see appenix at the en of the pape. We now tun to the poof of the main theoem of this section. u ɛ k u in C([, T ), L p ), (2.42) u ɛ k v ɛ k uv in C([, T ), L p ). (2.43) Poof of Theoem 6. Let φ C c (, T ) be a scala test function. It is obvious that u ɛ an v ɛ satisfy: T T u ɛ (t) φ (t) t + iv ( u ɛ (t) v ɛ (t) ) φ(t) t =, (2.44) v ɛ (t, x) = (u ɛ (t) K ɛ )(x) fo all (t, x) [, T ] R, (2.45) u ɛ () = u ɛ, (2.46) whee the integals in (2.44) ae the integal of a continuous function fom [, T ] to the Banach space W 1,p (R ). Recall that the injection i : L p (R ) W 1,p (R ) an the iffeentiation xi : L p (R ) W 1,p (R ) ae boune linea opeato. Theefoe (2.42) an (2.43) imply u ɛ k u in C([, T ], W 1,p (R )), iv [u ɛ k v ɛ k ] iv [uv] in C([, T ], W 1,p (R )), (2.47) hich is moe than enough to pass to the limit in elation (2.44). To pass to the limit in (2.45), it is enough to note that fo all (t, x) [, T ] R 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) 1

11 an finally it is tivial to pass to the limit in elation (2.46). Equation (2.44) means that the continuous function u(t) (continuous function with values in W 1,p (R )) satisfies (2.12) in the istibutional sense. But (2.12) implies that the istibutional eivative u (t) is itself a continuous function with value in W 1,p (R ). Theefoe u(t) is iffeentiable in the classical sense, i.e, it belongs to C 1 ([, T ], W 1,p (R )), an (2.12) is satisfie in the classical sense. We now tun to the poof of (2.15). The u ɛ s satisfies (2.19). Integating ove [, t], t < T, we get t u ɛ (t) p L = p uɛ p L (p 1) u ɛ (s, x) p iv v ɛ (s, x) xt. (2.49) p R Poposition 16 togethe with the geneal inequality ( ) f p g p L 1 2p f p 1 L + p g p 1 L f g p L p (2.5) implies that (u ɛ ) p, u p C([, T ], L 1 (R )), (2.51) (u ɛ k ) p u p C([, T ], L 1 (R )). (2.52) On the othe han, eplacing K by K in (2.48) we see ight away that Combining (2.51)-(2.54) we obtain So we can pass to the limit in (2.49) to obtain iv v, iv v ɛ C([, T ], L (R )), (2.53) iv v ɛ k iv v in C([, T ], L (R )). (2.54) u p iv v, (u ɛ ) p iv v ɛ C([, T ], L 1 (R )), (2.55) (u ɛ k ) p iv v ɛ k u p iv v in C([, T ], L 1 (R )). (2.56) t u(t) p L = u p p L (p 1) u(s, x) p iv v(s, x) xt. p R But (2.55) implies that the function t u(t, x) p iv v(t, x) x is continuous, theefoe the function R t u(t) p Lp is iffeentiable an satisfies (2.15). 2.5 Continuation an conseve popeties Theoem 17 (Continuation of solutions). The solution povie by Theoem 6 can be continue up to a time T max (, + ]. If T max < +, then lim t Tmax sup τ [,t] u(τ) L p = + Poof. The poof is stana. One just nees to use the continuity of the solution with espect to time. Theoem 18 (Consevation of mass/ secon moment). (i) Une the assumption of Theoem 6, an if we assume moeove that u L 1 (R ), then the solution u belongs to C([, T ], L 1 (R )) an satisfies u(t) L 1 = u L 1 fo all t [, T ]. (ii) Une the assumption of Theoem 6, an if we assume moeove that u has boune secon moment, then the secon moment of u(t) stays boune fo all t [, T ]. Poof. We just nee to evisit the poof of Poposition 16. Since u L 1 L p it is clea that u ɛ = J ɛ u u in L 1 (R ) L p (R ). (2.57) 11

12 Using convegences (2.57), (2.38), (2.39) an (2.4) we pove that u ɛ k u in C([, T ], L 1 L p ). (2.58) The poof is exactly the same than the one of Poposition 16. Since the aggegation equation is a consevation law, it is obvious that the smooth appoximates satisfy u ɛ (t) L 1 = u ɛ L 1. Using (2.58) we obtain u(t) L 1 = u L 1. We now tun to the poof of (ii). Since the smooth appoximates u ɛ have compact suppot, thei secon moment is clealy finite, an the following manipulation ae justifie: ( ) 1/2 ( ) 1/2 x 2 u ɛ (t, x)x = 2 x v ɛ u ɛ (x) 2 x 2 u ɛ (t, x)x v ɛ 2 u ɛ (t, x)x t R R R R ( ) 1/2 C x 2 u ɛ (t, x)x. (2.59) R Assume now that the secon moment of u is boune. A simple computation shows that if η ɛ is aially symmetic, then x 2 η ɛ = x 2 + secon moment of η ɛ. Theefoe x 2 u ɛ (x)x x 2 η ɛ (x) u (x)x R R x 2 u (x)x + x 2 η ɛ (x)x R R x 2 u (x)x + 1 fo ɛ small enough. (2.6) R Inequality (2.6) come fom the fact that the secon moment of η ɛ goes to as ɛ goes to. Estimate (2.59) togethe with (2.6) povie us with a unifom boun of the secon moment of the u ɛ (t) which only epens the secon moment of u. Since u ɛ conveges to u in L 1, we obviously have, fo a given R an t: x R x 2 u(t, x)x = lim ɛ x R x 2 u ɛ (t, x)x lim sup ɛ R x 2 u ɛ (t, x)x. Since R is abitay, this show that the secon moment of u(t, ) is boune fo all t fo which the solution exists. Combining Theoem 17 an 18 togethe with equality (2.15) we get: Theoem 19 (Global existence when K is boune fom above). Une the assumption of Theoem 6, an if we assume moeove that u L 1 (R ) an ess sup K < +, then the solution u exists fo all times (i.e.: T max = + ). Poof. Equality (2.15) can be witten t { u(t) p Lp} = (p 1) u(t, x) p (u(t) K)(x) x. (2.61) R Since K is boune fom above we have (u(s) K)(x) (ess sup K) u(s, x)x = (ess sup K) u L 1. R (2.62) Combining (2.61), (2.62) an Gonwall inequality gives u(t) p L p u p L p e(p 1)(ess sup K) u L 1 t, so the L p -nom can not blow-up in finite time which, because of Theoem 17, implies global existence. 12

13 3 Uniqueness of solutions in P 2 (R ) L p (R ) In this section we use an optimal tanspot agument to pove uniqueness of solutions in P 2 (R ) L p (R ), when K W 1,q fo 1 < q < an p its Höle conjugate. To o that, we shall follow the steps in [19], whee the authos exten the wok of Loepe [34] to pove, among othe, uniqueness of P 2 L -solutions of the aggegation equation when the inteaction potential K has a Lipschitz singulaity at the oigin. One can easily check that solutions of the aggegation equation constucte in pevious sections ae istibution solutions, i.e. they satisfy T ( ) ϕ (t, x) + v(t, x) ϕ(t, x) u(t, x) x t = ϕ(, x)u (x) x (3.63) t R N R N fo all ϕ C ([, T ) R N )). A function u(t, x) satisfying (3.63) is sai to be a istibution solution to the continuity equation (2.12) with the given velocity fiel v(t, x) an initial ata u (x). In fact, it is uniquely chaacteize by u(t, x) x = u (x) x B X t (B) fo all measuable set B R, see [1]. Hee X t : R R is the flow map associate with the velocity fiel v(t, x) an X t is its invese. In the optimal tanspot teminology this is equivalent to say that X t tanspots the measue u onto u(t) (u(t) = X t #u ). We ecall, fo the sake of completeness, [19, Theoem 2.4], whee seveal esults of [2, 35, 22, 1] ae put togethe. Theoem 2 ([19]). Let ρ 1 an ρ 2 be two pobability measues on R N, such that they ae absolutely continuous with espect to the Lebesgue measue an W 2 (ρ 1, ρ 2 ) <, an let ρ θ be an intepolation measue between ρ 1 an ρ 2, efine as in [34] by ρ θ = ((θ 1)T + (2 θ)i R N ) # ρ 1 (3.64) fo θ [1, 2], whee T is the optimal tanspot map between ρ 1 an ρ 2 ue to Benie s theoem [12] an I R N is the ientity map. Then thee exists a vecto fiel ν θ L 2 (R N, ρ θ x) such that i. θ ρ θ + iv(ρ θ ν θ ) = fo all θ [1, 2]. ii. ρ θ ν θ 2 x = W2 2 (ρ 1, ρ 2 ) fo all θ [1, 2]. R N iii. We have the L p -intepolation estimate ρ θ L p (R N ) max { } ρ 1 L p (R N ), ρ 2 L p (R N ) fo all θ [1, 2]. Hee, W 2 (f, g) is the Eucliean Wassestein istance between two pobability measues f, g P(R n ), { } 1/2 W 2 (f, g) = inf v x 2 Π(v, x), (3.65) Π Γ R n R n whee Π uns ove the set of joint pobability measues on R n R n with maginals f an g. Now we ae eay to pove the uniqueness of solutions to the aggegation equation. Theoem 21 (Uniqueness). Let u 1, u 2 be two boune solutions of equation (2.12) in the inteval [, T ] with initial ata u P 2 (R ) L p (R ), 1 < p < an assume 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) fo all t T. 13

14 Poof. Consie two chaacteistics flow maps, X 1 an X 2, such that u i = X i #u, i = 1, 2. Define the quantity Q(t) := 1 X 1 (t) X 2 (t) 2 u (x) x, (3.66) 2 R N Fom [19, Remak 2.3], we have W2 2 (u 1 (t), u 2 (t)) 2Q(t) which we now pove is zeo fo all times, implying that u 1 = u 2. Now, to see that Q(t) we compute the eivative of Q with espect to time. Q t = X 1 X 2, v 1 (x 1 ) v 2 (x 2 ) ρ (x)x R N = X 1 X 2, v 1 (x 1 ) v 1 (x 2 ) ρ (x)x + X 1 X 2, v 1 (x 2 ) v 2 (x 2 ) ρ (x)x R N R N whee the time vaiable has been omitte fo claity. The above agument is justifie because, ue to Lemma 7, the velocity fiel is C 1 an boune. Taking into account the Lipschitz popeties of v into the fist integal an using Höle inequality in the secon one, we can wite ( Q t CQ(t) + Q(t) 1 2 R N u 1 (X 2 (t, x)) u 2 (X 2 (t, x)) 2 ρ (x)x = CQ(t) + Q(t) 1 2 I(t) 1 2. (3.67) Now, in oe to estimate I(t), we use that the solutions ae constucte tanspoting the initial ata though thei flow maps, so we can wite it as I(t) = K (u 1 u 2 ) [X 2 (t, x)] 2 u (x) x = R N K (u 1 u 2 ) (x) 2 u 2 (x) x. R N Thus, taking an intepolation measue ρ θ between ρ 1 an ρ 2 an using Höle inequality an fist statement of Theoem 2 we can get a boun fo I(t) I(t) ( 2 1 R N ( 2 ) 2q ) 1/q K θ u θ u 2 (t) L p (3.68) 1 D 2 K (ν θ u θ ) 2 L 2qθ u 2(t) L p, (3.69) whee ν θ L 2 (R N, υ θ x) is a vecto fiel, as escibe in Theoem 2. Let us wok on the fist tem of the ight han sie. Using Young inequality, fo α such that q = 1/q + 1/α we obtain 2 1 D 2 K (ν θ u θ ) 2 L 2qθ 2 1 ) 1 2 D 2 K 2 L q ν θu θ 2 Lαθ. (3.7) Note that q (1, + ) implies α (1, 2). Theefoe we can use Höle inequality with conjugate exponents 2/(2 α) an 2/α to obtain ( ) 2/α ( ) (2 α)/α ( ) ν θ u θ 2 L = u α θ α/2 u θ α/2 ν θ α u θ α/(2 α) u θ ν θ 2 (3.71) α whence, since we can see fom simple algebaic manipulations with the exponents that 2 α = p, the conjugate of q, 2 2 ( ) D 2 K (ν θ u θ ) 2 L 2qθ D 2K 2 L u q θ L p u θ ν θ 2 θ. (3.72) 1 Theefoe, using statements (ii) an (iii) of Theoem 2 we obtain 1 I(t) u 2 L p max{ u 1 L p, u 2 L p} D 2 K 2 qw 2 2 (u 1, u 2 ) CQ(t). (3.73) 14

15 Finally, going back to (3.67) we see that Q t Q(t), whence, since Q() =, we can conclue Q(t) an thus u 1 = u 2. The limiting case p = is the one stuie in [19]. Remak 22. Note that in oe to make the above agument igoous, we nee the gaient of the kenel to be at least C 1 when estimating I. It is not the case hee, but we can still obtain the estimate using smooth appoximations. let us efine I ɛ (t) = K ɛ (u 1 u 2 ) [X 2 (t, x)] 2 u (x) x R N whee K ɛ = J ɛ K (see section 2.2). Since K ɛ conveges to K in L q, it is clea that K ɛ (u 1 u 2 ) conveges pointwise to K (u 1 u 2 ). Using the ominate convegence theoem togethe with the fact that K ɛ (u 1 u 2 ) L is unifomly boune we get that I ɛ (t) conveges to I(t) fo evey t (, T ). On the othe han, ue to the efinition of u θ we can wite the iffeence u 2 u 1 as the integal between 1 an 2 of θ u θ with espect to θ. Now, since the equation θ u θ + iv(u θ ν θ ) = is satisfie in the sense of istibution, an K ɛ Cc (R ), we can eplace θ u θ fo iv(ν θ u θ ) an pass the ivegence to the othe tem of the convolution, so that the equality 2 1 (D 2 K ɛ ν θ u θ )(x)θ = K ɛ (u 2 u 1 )(x) hols fo all x R. The est of the manipulations pefome above ae staight fowa with K ɛ. passing to the limit in (3.73) is easy since D 2 K ɛ conveges to D 2 K in L q. 4 Instantaneous mass concentation when K(x) = x In this section we consie the aggegation equation with an inteaction potential equal to x in a neighbohoo of the oigin an whose gaient is compactly suppote (o ecay exponentially fast at infinity). The Laplacian of this kin of potentials has a 1/ x singulaity at the oigin, theefoe K belongs to W 1,q (R ) 1 if an only if q [1, ). The Höle conjugate of is. Using the theoy evelope in section 2 an 3 we theefoe get local existence an uniqueness of solutions in P 2 (R ) L p (R ) fo all p > 1. Hee we stuy the case whee the initial ata is in P 2 (R ) L p (R ) fo p < 1. Given p < 1 we exhibit initial ata in P 2(R ) L p (R ) fo which the solution instantaneously concentates mass at the oigin (i.e. a elta Diac at the oigin is ceate instantaneously). This shows that the existence theoy evelope in section 2 an 3 is in some sense shap. This also shows that it is possible fo a solution to lose instantaneously its absolute continuity with espect to the Lebesgue measue. The solutions constucte in this section have compact suppot, hence we can simply consie K(x) = x without changing the behavio of the solution, given that if the solution has a small enough suppot, it only feels the pat of the potential aoun the oigin. We buil on the wok evelope in [16] on global existence fo measue solutions with boune secon moment: Theoem 23 (Existence an uniqueness of measue solutions [16]). Suppose K(x) = x. Given µ P 2 (R ), thee exists a unique weakly continuous family of pobability measues (µ t ) t (,+ ) satisfying t µ t + iv(µ t v t ) = in D ((, ) R ), (4.74) v t = K µ t, (4.75) µ t conveges weakly to µ as t. (4.76) 15

16 Hee K is the unique element of minimal nom in the subiffeential of K. Simply speaking, since K(x) = x is smooth away fom the oigin an aially symmetic, we have K(x) = x x fo x an K() =, an thus: ( x y K µ)(x) = µ(y). (4.77) x y Note that, µ t being a measue, it is impotant fo K to be efine fo evey x R so that (4.75) makes sense. Equation (4.74) means that + R y x ( ψ ) t (x, t) + ψ(x, t) v t(x) µ t (x) t =, (4.78) fo all ψ C (R (, + )). Fom (4.77) it is clea that v t (x) 1 fo all x an t, theefoe the above integal makes sense. The main Theoem of this section is the following: Theoem 24 (Instantaneous mass concentation). Consie the initial ata u (x) = { L x 1+ɛ if x < 1, othewise, (4.79) whee ɛ (, 1) an L := ( x <1 x ( 1+ɛ) x) 1 is a nomalizing constant. Note that u L p (R ) fo all p [1, 1+ɛ ). Let (µ t) t (,+ ) be the unique measue solution of the aggegation equation with inteaction potential K(x) = x an with initial ata u. Then, fo evey t > we have µ t ({}) >, i.e., mass is concentate at the oigin instantaneously an the solution is no longe continuous with espect to the Lebesgue measue. Theoem 24 is a consequence of the following estimate on the velocity fiel: Poposition 25. Let (µ t ) t (,+ ) be the unique measue solution of the aggegation equation with inteaction potential K(x) = x an with initial ata (4.79). Then, fo all t [, + ) the velocity fiel v t = K µ t is focussing an thee exists a constant C > such that v t (x) C x 1 ɛ fo all t [, + ) an x B(, 1). (4.8) By focussing, we mean that the velocity fiel points inwa, i.e. λ t : [, + ) [, + ) such that v t (x) = λ t ( x ) x x. thee exists a nonnegative function 4.1 Repesentation fomula fo aially symmetic measue solutions In this section, we show that fo aially symmetic measue solutions, the chaacteistics ae well efine. As a consequence, the solution to (2.12) can be expesse as the push fowa of the initial ata by the flow map associate with the ODE efining the chaacteistics. In the following the unit sphee {x R, x = 1} is enote by S an its suface aea by ω. Definition 26. If µ P(R ) is a aially symmetic pobability measue, then we efine ˆµ P([, + )) by fo all I B([, + )). ˆµ(I) = µ({x R : x I}) 16

17 Remak 27. If a measue µ is aially symmetic, then µ({x}) = fo all x, an theefoe K(x y) µ(y) = R \{x} K(x y) µ(y) R fo all x. In othe wos, fo x, ( K µ)(x) is well efine espite the fact that K is not efine at x =. As a consequence ( K µ)(x) = ( K µ)(x) if x an ( K µ)() =. Remak 28. If the aially symmetic measue µ is continuous with espect to the Lebesgue measue an has aially symmetic ensity u(x) = ũ( x ), then ˆµ is also continuous with espect to the Lebesgue measue an has ensity û, whee û() = ω 1 ũ(). (4.81) Lemma 29 (Pola cooinate fomula fo the convolution). Suppose µ P(R ) is aially symmetic. Let K(x) = x, then fo all x we have: ( + ( ) ) x x (µ K) (x) = φ ˆµ(ρ) (4.82) ρ x whee the function φ : [, + ) [ 1, 1] is efine by φ() = 1 e 1 y ω e 1 y e 1σ(y). (4.83) Poof. This come fom simple algebaic manipulations. These manipulations ae shown in [5]. In the next Lemma we state popeties of the function φ efine in (4.83). Lemma 3 (Popeties of the function φ). S (i) φ is continuous an non-eceasing on [, + ). Moeove φ() =, an lim φ() = 1. (ii) φ() is O() as. To be moe pecise: Poof. Consie the function F : [, + ) S [ 1, 1] efine by Since F is boune, we have that φ() = 1 ω φ() lim = 1 1 (y e 1 ) ω > S 2 σ(y). (4.84) (, y) e 1 y e 1 y e 1. (4.85) S F (, y)σ(y) = 1 ω S \{e 1} F (, y)σ(y). If y S \{e 1 } then the function F (, y) is continuous on [, + ) an C on (, + ). An explicit computation shows then that F 1 F (, y)2 (, y) =, (4.86) e 1 y thus φ is non-eceasing an, by the Lebesgue ominate convegence, it is easy to see that φ is continuous, φ() = an lim φ() = 1, which pove (i). To pove (ii), note that the function F (, y) can be extene by continuity on [, + ). Theefoe the ight eivative with espect to of F (, y) is well efine: F (, y) F (, y) 1 F (, y)2 lim = = 1 (y e 1 ) 2. y > 17

18 an since F/ is boune on (, + ) S \{e 1 }, we can now use the Lebesgue ominate convegence theoem to conclue: φ() φ() φ() 1 F (, y) F (, y) lim = lim = lim σ(y) = 1 1 (y e 1 ) ω > > > S ω S 2 σ(y). ( Remak 31. Note that fo > the function ρ φ ρ ) is non inceasing an continuous. Inee, it is equal to 1 when ρ = an it eceases to as ρ. In paticula, the integal in (4.82) is well efine fo any pobability measue ˆµ P([, + )). Remak 32. In imension two, it is easy to check that lim 1 φ () = + which implies that the eivative of the function φ has a singulaity at = 1 an thus, that the function φ is not C 1. Poposition 33 (Chaacteistic ODE). Let K(x) = x an let (µ t ) t [,+ ) be a weakly continuous family of aially symmetic pobability measues. Then the velocity fiel { ( K µ t )(x) if x v(x, t) = (4.87) if x = is continuous on R \{} [ + ). Moeove, fo evey x R, thee exists an absolutely continuous function t X t (x), t [, + ), which satisfies t X t(x) = v(x t (x), t) fo a.e. t (, + ), (4.88) X (x) = x. (4.89) Poof. Fom fomula (4.82), Remak 31, an the weak continuity of the family (µ t ) t [,+ ), we obtain continuity in time. The continuity in space simply comes fom the continuity an bouneness of the function φ togethe with the Lebesgue ominate convegence theoem. Since v is continuous on R \{} [ + ) we know fom the Peano theoem that given x R \{}, the initial value poblem (4.88)-(4.89) has a C 1 solution at least fo shot time. We want to see that it is efine fo all time. Fo that, note that by a continuation agument, the inteval given by Peano theoem can be extene as long as the solution stays in R \{}. Then, if we enote by T x the maximum time so that the solution exists in [, T x ) we have that eithe T x = an we ae one, o T x < +, in which case clealy lim t Tx X t (x) =, an we can exten the function X t (x) on [, + ) by setting X t (x) := fo t T x. The function t X t (x) that we have just constucte is continuous on [, + ), C 1 on [, + )\{T x } an satisfies (4.88) on [, + )\{T x }. If x =, we obviously let X t (x) = fo all t. Finally, we pesent the epesentation fomula, by which we expess the solution to (2.12) as a pushfowa of the initial ata. See [1] o [41] fo a efinition of the push-fowa of a measue by a map. Poposition 34 (Repesentation fomula). Let (µ t ) t [,+ ) be a aially symmetic measue solution of the aggegation equation with inteaction potential K(x) = x, an let X t : R R be efine by (4.87), (4.88) an (4.89). Then fo all t, µ t = X t #µ. Poof. In this poof, we follow aguments fom [1]. Since fo a given x the function t X t (x) is continuous, one can easily pove, using the Lebesgue ominate convegence theoem, that t X t #µ is weakly continuous. Let us now pove that µ t := X t #µ satisfies (4.78) fo all ψ C (R (, )). Given that the test function ψ is compactly suppote, thee exist T > such that ψ(x, t) = fo all t T. We theefoe have: ( ) = ψ(x, T )µ T (x) ψ(x, )µ (x) = ψ(x T (x), T ) ψ(x, ) µ (x). (4.9) R R R 18

19 If we now take into account that fom Poposition 33 the mapping t φ(x t (x), t) is absolutely continuous, we can ewite (4.9) as = = = = T R T R T T R R ( t ψ(x t(x), t) ) t µ (x) (4.91) ( ψ(x t (x), t) v(x t (x), t) + ψ t (X t(x), t) ( ψ(x t (x), t) v(x t (x), t) + ψ t (X t(x), t) ( ψ(x, t) v(x, t) + ψ ) t (x, t) µ t (x) t. ) t µ (x) (4.92) ) µ (x) t (4.93) The step fom (4.92) to (4.93) hols because of the fact that v(x, t) 1, which justifies the use of the Fubini Theoem. Remak 35 (Repesentation fomula in pola cooinates). Let µ t an X t be as in the pevious poposition. Let R t : [, + ) [, + ) be the function such that X t (x) = R t ( x ). Then ˆµ t = R t #ˆµ. (4.94) Remak 36. Since φ is nonnegative (Lemma 3), fom (4.82), (4.87) an (4.88) we see that the function t X t (x) = R t ( x ) is non inceasing. 4.2 Poof of Poposition 25 an Theoem 24 We ae now eay to pove the estimate on the velocity fiel an the instantaneous concentation esult. We stat by giving a fozen in time estimate of the velocity fiel. Lemma 37. Let K(x) = x, an let u (x) be efine by (4.79) fo some ɛ (, 1). Then thee exist a constant C > such that (u K) (x) C x 1 ɛ (4.95) fo all x B(, 1)\{}. Poof. Note that if we o the change of vaiable s = x ρ (u K) (x) = x in equation (4.82), we fin that + φ(s) û ( x s )s s 2. (4.96) On the othe han, using (4.81) we see that the û () coesponing to the u (x) efine by (4.79) is û () = { ω ɛ if < 1, othewise. (4.97) Then, plugging (4.97) in (4.96) we obtain that fo all x (u K) (x) = ω x 1 ɛ + x φ(ρ) ρ. (4.98) ρ2 ɛ In light of statement (ii) of Lemma 3, we see that the pevious integal conveges as x. (u K) (x) is O( x 1 ɛ ) as x an (4.95) follows. Hence Finally, the last piece we nee in oe to pove Poposition 25 fom the pevious lemma, is the following compaison pinciple: 19

20 Lemma 38 (Tempoal monotonicity of the velocity). Let (µ t ) t (,+ ) be a aially symmetic measue solution of the aggegation equation with inteaction potential K(x) = x. Then, fo evey x R \{} the function t ( K µ t )(x) is non eceasing. Poof. Combining (4.82) an (4.94) we see that (µ t K)(x) = ( + ) φ( x R t (ρ) ) ˆµ (ρ). (4.99) Now, by Lemma 3, φ is non eceasing an ue to Remak 35, t R t (ρ) is non inceasing. Hencefoth it is clea that (4.99) is itself non eceasing. At this point, Poposition 25 follows as a simple consequence of the fozen in time estimate (4.95) togethe with Lemma 38, an we can give an easy poof fo the main esult we intouce at the beginning of the section. Poof of Theoem 24. Using the epesentation fomula (Poposition 34) an the efinition of the push fowa we get µ t ({}) = (X t #µ )({}) = µ (Xt 1 ({})). Then, note that the solution of the ODE ṙ = 1 ɛ eaches zeo in finite time. Theefoe, fom Poposition 25 an 33 we obtain that fo all t >, thee exists δ > such that X t (x) = fo all x < δ. In othe wos, fo all t >, thee exists δ > such that B(, δ) Xt 1 ({}). Clealy, given ou choice of initial conition, we have that µ (B(, δ)) > if δ >, an theefoe µ t ({}) > if t >. 5 Osgoo conition fo global well-poseness This section consies the global well-poseness of the aggegation equation in P 2 (R ) L p (R ), epening on the potential K. We stat by giving a pecise efinitions of natual potential, epulsive in the shot ange an stictly attactive in the shot ange, an then we pove Theoem 5. Definition 39. A natual potential is a aially symmetic potential K(x) = k( x ), whee k : (, + ) R is a smooth function which satisfies the following conitions: (C1) sup k () < +, (, ) (C2) α > such that k () an k () ae O(1/ α ) as +, (MN1) δ 1 > such that k () is monotonic (eithe inceasing o eceasing) in (, δ 1 ), (MN2) δ 2 > such that k () is monotonic (eithe inceasing o eceasing) in (, δ 2 ). Remak 4. Note that monotonicity conition (MN1) implies that k () an k() ae also monotonic in some (iffeent) neighbohoo of the oigin (, δ). Also, note that (C1) an (MN1) imply (C3) lim k () exists an is finite. + Remak 41. The fa fiel conition (C2) can be oppe when the ata has compact suppot. 2

21 Definition 42. A natual potential is sai to be epulsive in the shot ange if thee exists an inteval (, δ) on which k() is eceasing. A natual potential is sai to be stictly attactive in the shot ange if thee exists an inteval (, δ) on which k() is stictly inceasing. We woul like to emak that the two monotonicity conitions ae not vey estictive as, in oe to violate them, a potential woul have to exhibit some pathological behavio aoun the oigin, like oscillating faste an faste as. 5.1 Popeties of natual potentials As a last step befoe poving Theoem 5, let us point out some popeties of natual potentials, which show the eason behin the choice of this kin of potentials to wok with. Lemma 43. If K(x) = k( x ) is a natual potential, then k () = o(1/) as. Poof. Fist, note that since k() is smooth away fom we have k (1) k (ɛ) = 1 ɛ k (). Now, because of (MN1) we know that thee exists a neighbohoo of zeo in which k oesn t change sign. Theefoe, letting ɛ an using (C3) we conclue that k is integable aoun the oigin. A simple integation by pat, togethe with (C3) gives then that k (s)s s = Diviing both sies by an letting we obtain 1 lim which, combine with (MN2) implies lim k () =. k (s) + k (). k (s)s s =. The next lemma shows that the existence theoy evelope in the pevious section applies to this class of potentials. Lemma 44. If K is a natual potential then K W 1,q (R ) fo all 1 q <. As a consequence, the citical exponents p s an q s associate to a natual potential satisfy Poof. Recall that K(x) = k ( x ) x x an q s an p s 1. 2 ( ) K (x) = k ( x ) k ( x ) xi x j x i x j x x 2 + δ ij k ( x ), x whee δ ij is the Konecke elta symbol. In oe to pove the lemma, it is enough to show that k ( x ), k ( x ) an k ( x ) x belong to L q (R ) fo all 1 q <. To o that, obseve that the ecay conition (C2) implies that they belong to L q (B(, 1) c ) fo all q 1. Then, we take into account that (C3) implies that k () = O(1/) as an that we have seen in the pevious Lemma that k () = o(1/) as. This is enough to conclue, since the function x 1/ x is in L q (B(, 1)) fo all 1 q <. The following Lemma togethe with Theoem 19 gives global existence of solutions fo natual potentials which ae epulsive in the shot ange, whence pat (i) of Theoem 5 follows. 21

22 Lemma 45. Suppose K is a natual potential which is epulsive in the shot ange. Then K is boune fom above. Poof. We will pove that thee is a neighbohoo of zeo on which K. This combine with the ecay conition (C2) give the esie esult. Fist, ecall that K(x) = k ( x ) + ( 1)k ( x ) x 1. Then, since k is epulsive in the shot ange, thee exists a neighbohoo of zeo in which k. Now, we have two possibilities: on one han, if lim + k () =, then given (, + ) thee exists s (, ) such that k () = k (s). Togethe with (MN1), this implies that k is also non-positive in some neighbohoo of zeo. On the othe han if lim + k () <, then the fact that k () = o(1/) implies that k ()+( 1)k () is negative fo small enough. Finally, the next Lemma will be neee to pove global existence fo natual potentials which ae stictly attactive in the shot ange an satisfy the Osgoo citeia. Lemma 46. Suppose that K is a natual potential which is stictly attactive in the shot ange an satisfies the Osgoo citeia (1.9). If moeove sup x K(x) = + then the following hols (Z1) lim + k () = + an lim + k () = +, (Z2) δ 1 > such that k () an k () ae eceasing fo (, δ 1 ), (Z3) δ 2 > such that k () k () fo (, δ 2 ). Poof. Let us stat by poving by contaiction that If we suppose that lim sup + p (,) k () k () = +. (5.1) < C (, 1], (5.11) then given a sequence n + thee will exist anothe sequence s n +, < s n < n, such that k ( n ) n = k (s n ) < C. Since k () is monotonic aoun zeo, this implies that k is boune fom above, an combining this with (5.11) we see that K must also be boune fom above, which contaicts ou assumption. Now, statements (Z1), (Z2), (Z3) follow easily: Fist note that if lim + k () >, then clealy the Osgoo conition (1.9) is not satisfie, whence lim + k () =. This implies that fo all > thee exists s (, ) such that k () = k (s). (5.12) Combining (5.12), (5.1) an the monotonicity of k we get that lim + k () = + an k is eceasing on some inteval (, δ), which coespons with the fist pat of (Z1) an (Z2). Now, going back to (5.12) we see that if < s < < δ then k () = k (s) k () which poves (Z3). This implies { k } () = 1 ( ) k () k () an theefoe k ()/ eceases on (, δ). Thus, (5.1) implies lim + k () complete. = + an the poof is 22