A Kinetic Model for the formation of Swarms with nonlinear interactions

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1 A Knetc Model for the formaton of Swarms wth nonlnear nteractons Martn Parsot, Mroslaw Lachowcz To cte ths verson: Martn Parsot, Mroslaw Lachowcz. A Knetc Model for the formaton of Swarms wth nonlnear nteractons. Knetc and Related Models, AIMS, 26, 9, pp.33. < <.3934/krm >. <hal-52397> HAL Id: hal Submtted on 6 May 25 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 Ttle: A Knetc Model for the formaton of Swarms wth nonlnear nteractons Short ttle: Nonlnear Knetc Model for the formaton of Swarm Correspondng author: martn.parsot@nra.fr Abstract: The present paper deals wth the modelng of formaton and destructon of swarms usng a nonlnear Boltzmann lke equaton. We ntroduce a new model that contans parameters characterzng the attractveness or repulsveness of ndvduals. The model can represent both gregarous and soltarous behavors. In the latter case we provde a mathematcal analyss n the space homogeneous case. Moreover we dentfy relevant hydrodynamc lmts on a formal way. We ntroduce some prelmnary results n the case of gregarous behavor and we ndcate open problems for further research. Fnally, we provde numercal smulatons to llustrate the ablty of the model to represent formaton or destructon of swarms.

3 Manuscrpt submtted to AIMS Journals olume X, Number X, XX 2X do:.3934/xx.xx.xx.xx pp. X XX A KINETIC MODEL FOR THE FORMATION OF SWARMS WITH NONLINEAR INTERACTIONS Martn Parsot,2,3,4 and Mros Law Lachowcz 5,6 INRIA, ANGE Project-Team, Rocquencourt, F-7853 Le Chesnay Cedex, France 2 CEREMA, F-628 Margny-Lès-Compègne, France 3 CNRS, UMR 7598, Laboratore Jacques-Lous Lons, F-755, Pars, France 4 Sorbonne Unverstés, UPMC Unv Pars 6, UMR 7598, Laboratore Jacques-Lous Lons, F- 755, Pars, France 5 Insttute of Appled Mathematcs and Mechancs, Faculty of Mathematcs, Informatcs and Mechancs, Unversty of Warsaw, ul. Banacha 2, 2-97 Warsaw, Poland 6 Honorary Professor, School of Mathematcs, Statstcs and Computer Scence, Unversty of KwaZulu Natal, South Afrca Communcated by the assocate edtor name Abstract. The present paper deals wth the modelng of formaton and destructon of swarms usng a nonlnear Boltzmann lke equaton. We ntroduce a new model that contans parameters characterzng the attractveness or repulsveness of ndvduals. The model can represent both gregarous and soltarous behavors. In the latter case we provde a mathematcal analyss n the space homogeneous case. Moreover we dentfy relevant hydrodynamc lmts on a formal way. We ntroduce some prelmnary results n the case of gregarous behavor and we ndcate open problems for further research. Fnally, we provde numercal smulatons to llustrate the ablty of the model to represent formaton or destructon of swarms.. Introducton. We are gong to model the swarmng behavor of an ndvdual populaton. Let f t, x, v be a probablty densty of ndvduals at tme t and poston x R d wth velocty v R d. We assume that, the set of veloctes of the ndvduals, s compact. We model the evoluton of populatons at the mesoscopc scale by the nonlnear ntegro dfferental Boltzmann lke equaton: t f t, x, v + v x f t, x, v = Q [f] t, x, v ε = T [f t, x,.] w, v f t, x, w T [f t, x,.] v, w f t, x, v dw ε 2 Mathematcs Subject Classfcaton. 47G2, 76N, 78M35, 82C22, 92C5, 92D25. Key words and phrases. knetc formulaton; collectve behavour; swarm; self-organsaton; orentaton nteracton; hydrodynamc lmt. M.L. acknowledges a support from the Unversty of KwaZulu Natal Research Found South Afrca. The work of M.P. was carred out durng the tenure of an ERCIM Alan Bensoussan Fellowshp Programme. The research leadng to these results has receved fundng from the European Unon Seventh Framework Programme FP7/27-23 under grant agreement n o

4 NONLINEAR KINETIC MODEL FOR THE FORMATION OF SWARM 3 wth the ntal data f, x, v = F x, v and the parameter ε that corresponds to the Knudsen number, see [25]. The operator Q descrbes nteractons between ndvduals. We only consder local n space nteractons that s physcally justfed at least for large characterstc space scale. Several models of swarmng wth the long range nteractons descrbed by the mean feld approach, can be found n the lterature, see for example [, 23, 7]. The turnng rate T [f] v, w measures the probablty for an ndvdual wth velocty v to change velocty nto w. Lnear turnng rates wth nfluence of the orentaton of ndvduals were analyzed n detals n [6, 2]. In ths paper we focus on the followng general nonlnear case T [f t, x,.] v, w = σ ρ,x β ρ,x v, w f t, x, w γρ,x 2 wth the macroscopc densty of ndvduals ρ t, x = f t, x, v dv. For any ρ and at any x, the nteracton rate β ρ,x : 2 R +, the attractveness coeffcent γ ρ,x R +, and σ ρ,x {, } characterze the nteracton between the ndvdual agents. The nteracton rate β ρ,x corresponds to the tendency of ndvduals to swtch to a dfferent velocty. In the followng we assume ths nteracton rate s symmetrc, postve, bounded and separated from zero. The attractveness coeffcent γ ρ,x corresponds to the attractveness or repulsveness of ndvduals. Accordngly to observatons, the attractveness depends on the sze of the populaton [32], on the local resources [3] or even morphologcal adaptaton [7]. Note that the bologcally relevant nteractons are such that σ ρ,x =. However, n the present paper we are gong to study also the negatve nteractons,.e. σ ρ,x =, because of nterestng mathematcal propertes. A smpler framework wth only two possble veloctes +, was consdered n [, 2]. For smplcty of notaton, we do not ndcate the densty and the poston dependence of the collson parameters, as long as ths dependence s obvous. The swarm behavor, for some populatons also called herds, flocks, packs, schools, or shoals, s referred to as the self organzaton of ndvdual agents. There exsts a huge lterature related to swarm phenomena. Here we menton few examples and refer to the bblography theren. Ref. [3] revews hyperbolc and knetc models for self organzed bologcal aggregatons and traffc lke movement. Book [26] provdes the mathematcal modelng based on a mesoscopc descrpton and the constructon of effcent smulaton algorthms by Monte Carlo methods for collectve phenomena and self organzaton n systems composed of large numbers of ndvduals. Paper [24] deals wth plasma knetc theory to derve the correspondng hydrodynamc equaton for the densty of Daphncle. An nterestng agent-based stochastc model of vortex swarmng n Daphna has been proposed n [22]. A cell based model has been consdered n [35] and the effect of socal nteractons between cells has been descrbed. The model quantfes the contrbuton of ndvdual motlty engnes to swarmng. Reference [36] deals wth mechanstc modelng of swarms, the propertes of swarm models and the correspondng numercal algorthms. In paper [8] a numercal scheme has been developed to estmate coeffcents n nonlnear advecton dffuson equatons from ndvdual based model smulatons. The bophyscal prncples that cause the Proteus mrabls the swarm phenomena are gven n [5]. A swarmng model on a two dmensonal lattce, where the self propelled partcles exhbt a tendency to algn ferromagnetcally, has been studed n [27]. Paper [28] has shown that the transton to collectve moton n colones of gldng bacteral cells confned to a monolayer appears through the organzaton of

5 4 PARISOT MARTIN AND LACHOWICZ MIROS LAW cells nto larger movng clusters. The possble approaches for swarmng behavor n locusts have been descrbed and dscussed n [2]. The paper [4] deals wth a stochastc ndvdual based models related to the locusts behavor. In paper [9] the Cucker Smale partcle model, [8], for flockng has been dscussed and condtons for flockng have been stated. A lasov type knetc model has been derved and tme asymptotc flockng behavor has been proved. The asymptotc behavor of solutons of the Cucker Smale model has been studed n [6]. Reference [5] presents the knetc theory approach for swarmng systems of nteractng, self propelled dscrete partcles. The related macroscopc hydrodynamc equatons are derved from the Louvlle equaton. General solutons nclude flocks of constant densty and fxed velocty and other non trval morphologes.. entropy dsspaton law 2. global exstence maxmum prncple 3. dffuson equaton. self organzaton 2. global exstence vanshng n fnte tme 3. transport equaton γ. self organzaton 2. unqueness blow-up? open problem 3. transport equaton. entropy dsspaton law 2. global exstence maxmum prncple 3. dffuson equaton σ = σ = Fgure. Propertes accordngly to the parameters σ and γ.. classfcaton of the evoluton system 2. man propertes n the space homogeneous case 3. formal hydrodynamc lmt The present paper deals wth the mathematcal and numercal study of the orentatonal aggregaton. The man novelty wth respect to the prevous studes [6, 2] s that we consder non lnear turnng rate characterzed by the coeffcent of attractveness whch control the behavor of the solutons. In order to smplfy the study, we consder that ndvduals drectly swtch to ther objectve,.e. usng the notaton defned n [2] n the case of determnstc turnng G σ = δ Drac functon and wth optmal reorentaton M w v = w. The paper s organzed as follow. In Secton 2, we perform a mathematcal analyss of Eq. n the space homogeneous case. The man results of ths secton are summarzed n Fgure. We wll see that the behavor of the solutons s manly controlled by the sgn of σ γ. In the case of σ γ >, the evoluton of the dstrbuton functon satsfes an entropy dsspaton law and a maxmum prncple n the velocty doman. From the bologcal pont of vew, ndvduals attempt to dsperse and such a behavor s called soltarous, see [32, 3, 7]. In the case of σ γ < the evoluton of the dstrbuton functon s self organzed,.e. ndvduals attempt to algn themselves along at least one drecton. Ths case corresponds to the so called gregarous behavor, cf. [32, 3, 7]. The man open questons concern ths last case. We are currently not able to predct eventual blow up n fnte tme for postve self organzed nteractons. In Secton 3, we formally descrbe the spacal hydrodynamc lmts n the space dependent case for the two behavors. For ths purpose, we assume that the man

6 NONLINEAR KINETIC MODEL FOR THE FORMATION OF SWARM 5 term of dstrbuton, so called local equlbrum, corresponds to the stable steady state of the homogeneous case. In Secton 3. we focus on the nteractons satsfyng the dsspaton law. We show that the only stable steady state s the constant functon n the whole velocty doman, the so called dffusve pcture. The formal hydrodynamc lmt leads to a parabolc equaton wth a nonlnear coeffcent proportonal to ρ γ characterstc to the Carleman lke model, see [3]. We estmate the parameters of the macroscopc model, namely the dffuson parameter for some gven velocty doman. The mathematcal dervaton of an hydrodynamc lmt of our model for self organzed nteractons s currently an open problem. However, we propose a conjectured soluton, namely the algned pcture, based on the asymptotc soluton of the space homogeneous equaton and n the case of constant nteracton rate β. Fnally n Secton 4 we present varous numercal strateges to solve Eq.. Frst, we focus on the nteracton operator n Secton 4.2. In order to numercally observe the dfferent solutons n the non Lpschtzan case γ <, we propose several scheme based on lnear and nonlnear approxmaton of the space homogeneous equaton. Numercal smulatons llustratng the results are presented n Secton 5.. Then, we propose a complete dscretzaton of the PDE. The conjectured soluton n the gregarous case, namely the algned pcture, s not recover usng classcal transport scheme due to numercal dffuson. We propose n Secton 4.3 an ant dffusve scheme for Cartesan grd able to recover the algned pcture and we perform numercal smulatons n Secton Mathematcal analyss of space homogeneous case. In ths secton we provde the mathematcal analyss of Eq. wth 2 n the space homogeneous case,.e. when all functons are x ndependent. In addton, we smplfy the study by consderng restrctve assumptons on the nteracton rate Assumpton. We assume that β s a postve, symmetrc and bounded functon separated from zero,.e. there exsts < µ such that for almost any v, w 2, we have µ β L 2 β v, w = β w, v β L 2. The followng a pror conservaton propertes hold Theorem 2.. The total densty of ndvduals s a pror preserved n tme,.e. t ρ dx = then ρ t, x dx = ρ, x dx. R d R d R d Proposton. The operator Q s homogeneous of degree γ,.e. [ ] f Q [f] t, x, v = ρ γ Q t, x, v. ρ Wthout the lost of generalty, we may scale the soluton such that the macroscopc densty of the scaled soluton s,.e. g L =. The space homogeneous verson of Eq. reads εσ t g = β g g γ β g γ g wth g, v = G v 3 where s the convoluton product n the velocty doman,.e. β φ v = β v, w φ w dw.

7 6 PARISOT MARTIN AND LACHOWICZ MIROS LAW Note that we can return to the orgnal varable by settng f t, x, v = ρg ρ γ t, v. In ths secton we set ε = for smplcty. Proposton 2. Let η : R + R be a bounded convex dfferentable functon. The soluton g of Eq. 3 a pror satsfes { d, σ γ > entropy dsspaton law η g dv dt, σ γ < self-organzaton. Proof. To smplfy the notaton, we set g v = g t, v and g w = g t, w. By drect estmaton we have d η g v dv = σ β v, w gv γ g w g dt wg γ v η g v dw dv = σ β v, w g v g w g γ v gw γ η g v dw dv. Exchangng v and w and usng the symmetrc property of β v, w we obtan d η g v dv = σ β v, w g v g w g γ v g γ w η g v η g w dw dv. dt 2 Then the concluson follows snce the applcaton x x γ s decreasng for γ and ncreasng for γ. Accordng to Proposton 2, the solutons of Eq. 3 have completely dfferent evoluton for dfferent sgns of σ γ. In the followng, we are gong to analyze Eq. 3 consderng separately each doman of parameters. Note that γ = leads to the vanshng nteractons and to the trval soluton g t, v = G v. More generally, the non homogeneous soluton s n ths case smply shfted wth the velocty v, and we have f t, x, v = F x vt, v. From now on, we assume γ. We are only nterested n postve solutons of Eq. 3 for physcal reasons. We denote by L p +, for < p, the set of nonnegatve functons n the L p -space,.e. f L p + ff f L p and f. 2.. Entropy dsspatve nteractons: soltarous behavor Case σ = and < γ. In ths secton we consder Eq. 3 wth σ = and < γ. Theorem 2.2. Let σ =, < γ and β satsfy Assumpton. Assume G L + be such that there exst nonnegatve constants m M < + satsfyng m G v M, for almost any v. Then there exsts an unque global soluton g C R + ; L of Eq. 3 wth g, v = G v. Moreover, the soluton g satsfes the maxmum prncple,.e. for any t > and almost any v. m g t, v M,

8 NONLINEAR KINETIC MODEL FOR THE FORMATION OF SWARM 7 Proof. Consder the auxlary ODE problem defned by φ = p qφ α wth φ = φ, 4 α > and p and q are gven contnuous functons on R +. Accordng to the Pcard Lndelöf theorem, there exsts an unque local soluton and t can be prolonged to R +. In addton φ s nonnegatve when both p and the ntal data φ are nonnegatve. We approxmate the soluton of Eq. 3 by the sequence startng wth g t, v = G v and g n+ t, v defned as the unque soluton of the auxlary problem 4 wth α = γ, p = β g γ n g n, q = β g n and the ntal data φ = G v for any fxed v. By the propertes of the auxlary problem 4 we deduce that the sequence g n n s well defned. In addton, the g satsfes the maxmum prncple snce t s constant n tme. Let us now assume that the bounds are verfed by g n,.e. m g n t, v M, for any t, v R +. Then we consder the next approxmaton g n+. Usng the postve part functon 2 φ + = φ + φ, we wrte g n+ M + t g n+ M = β g γ n g n β g n M γ g n+ M + β g n g γ n+ M γ g n+ M +. We have the followng nequalty for < γ β gn γ = β v, w g n w gn γ w dw β v, w g n w dw M γ = β g n M γ. It follows β g γ n β g n M γ. Snce the functon x x γ s ncreasng and g n s nonnegatve, the RHS s nonpostve. Wth smlar arguments, we treat the lower bound. The fnal step of the proof conssts n the convergence of the sequence g n n. We compare the tme dervatve of two consecutve approxmatons and we multply by the sgn of the dfference,.e. sgn g n+ g n t g n+ g n = 2 sgn g n+ g n β gn γ gn γ gn + g n + β gn γ + g γ n gn g n sgn g n+ g n β g n g n gn γ + β g n g γ n+ gγ n Snce γ the last term s nonpostve: sgn g n+ g n β g n g γ n+ gγ n. In addton, the functon g γ n s γm γ Lpschtz contnuous. Then we ntegrate over the velocty doman and we obtan t g n+ g n L γ + 2 M γ β g n g n L, wth β = β v, w L L = β v, w L dw dv. We conclude the convergence by classcal contracton arguments. In fact, for any t T, we have γ + 2 M γ βt n g n+ g n L t g g L,T ;L n!. It s clear that the RHS vanshes when n goes to nfnty.

9 8 PARISOT MARTIN AND LACHOWICZ MIROS LAW Case σ = and γ <. In ths secton we consder Eq. 3 for σ = and γ <. We show that the result of Theorem 2.2 stll holds. Theorem 2.3. Let σ =, γ < and β satsfy Assumpton. Assume G L + be such that there exst nonnegatve constants < m M < + satsfyng m G v M, for almost any v. Then there exsts an unque global soluton g C R + ; L of Eq. 3 wth g, v = G v. Moreover, the soluton g satsfes for any t > and almost any v. m g t, v M, Proof. We approxmate the soluton of Eq. 3 by the sequence startng from g t, v = G v and g n+ t, v s the unque soluton of the auxlary problem 4 wth α =, p = β g n gn, γ q = β gn γ and the ntal data φ = G v for any fxed v. We deduce from the auxlary problem 4 that the sequence g n n s well defned. Then we show the maxmum prncple wth the same strategy than the proof of Theorem 2.2 wth the followng nequalty for γ < β g n = β w v gn γ v gn γ v dw β w v gn γ v v dw M γ = β gn γ M γ. We conclude followng the proof of Theorem 2.2. Note that we consdered ntal data separated from zero,.e. < m. The case not separated from zero s more dffcult to treat snce the RHS s not Lpschtz contnuous for γ <. The results related to the sequence g n the exstence and the maxmum prncple could be proved as well. Unfortunately, we cannot prove the convergence of the sequence. In fact, for ntal data vanshng n some ponts, σ = and γ <, the soluton of Eq. 3 s not unque. Let v such that G v =, we can easly show that for any tme t, there exsts a soluton g of Eq. 3 satsfyng { =, f t t g t, v >, otherwse. Remark Bologcal context. The soltarous case corresponds to repulsve nteractons between ndvduals. In ths context, the ndvduals try to occupy the whole doman. It leads that the relevant soluton s the only soluton strctly postve for any tme strctly postve Self organzed nteractons: gregarous behavor Case σ = and γ <. In ths secton, we consder Eq. 3 for σ = and γ <. Theorem 2.4. Let σ =, γ <, β satsfy Assumpton and G L +.

10 NONLINEAR KINETIC MODEL FOR THE FORMATION OF SWARM 9 Then there exsts a soluton of Eq. 3 n C R + ; L and t s unque among the nonnegatve functons. Moreover, the soluton vanshes n a fnte tme on a subdoman of. In fact, for any v and t R + such that γ ln G γ L γ µ µ G γ G v < G γ and t L γ γ β L 2 G γ, L γ we have g t, v =. Proof. Consder the followng auxlary problem φ = qφ pφ γ, wth φ = φ, 5 p and q are gven non negatve contnuous functons on R +. Accordng to the Pcard Lndelöf theorem, there exts an unque local soluton of Eq. 5 n the neghborhood of any tme t such that φ t >. For small enough ntal data, the soluton s not unque n general. More precsely, for some attractveness coeffcents γ, there exst solutons of the auxlary problem Eq. 5, whch are not nonnegatve. For example, wth β v, w = and parameters γ = 2 3, p and q not dependng of tme, the followng functon p φ t = q + φ 3 p 3 e q3 t q 3, s a soluton of the auxlary problem and for φ < q s ntally postve and become negatve for a tme large enough. However, the vanshng functon s a trval soluton of the auxlary problem wth vanshng ntal data. Then, for not vanshng ntal data, we can extend the local soluton φ on R + by zero from the tme that t vanshes,.e. we defne {, f there exsts s t such that φ s = ψ t = φ t, otherwse. One can check that ψ t s a nonnegatve soluton of Eq. 5 n C R +. In γ addton, for the soluton nonnegatve and small enough,.e. when φ <, the soluton s non ncreasng,.e. φ. We conclude that the soluton ψ s the unque global soluton n the set of nonnegatve functons of Eq. 5 wth small γ ntal data,.e. φ <. p q For ntal data large enough φ p p q p γ q the soluton s ncreasng. We conclude that the soluton s unque snce the RHS s Lpschtz contnuous. However, t s not globally defned snce t can blow up n a fnte tme. For parameters p and q that do not depend on tme, an explct soluton of Eq. 5 can be gven by consderng the ODE satsfes by φ γ. We have φ t = wth τ = p q + φ γ p q e γqt γ, for t < τ, for t τ ln q p φ γ γq, f φ < p γ q +, otherwse.

11 PARISOT MARTIN AND LACHOWICZ MIROS LAW The non negatve soluton wth parameters p and q not dependng on tme s globally defned n R + for any nonnegatve ntal data. We approxmate the soluton of Eq. 3 by the sequence startng from g t, v = G v and g n+ t, v beng the unque non negatve soluton of the auxlary problem 5 wth p = β g n, q = β gn γ and the ntal data φ = G v for any fxed v. Now assume that g n L R + ; L + whch s true for g t, v = G v. We have t g γ n+ γ β L 2 g n γ L γ g γ n+ µ. Usng the Grönwall lemma, we conclude that g γ n+ L R + ; L +, then g n+ L R + ; L +. Snce g n t, v s nonnegatve, we show the convergence of the sequence n L R + ; L + usng the classcal contracton arguments. Snce the functon x x γ s concave and by Proposton 2, g L γ decreases. We have t g = β g γ g β g g γ β L 2 g γ L γ g µgγ β L 2 G γ L γ g µgγ. It follows that the soluton of the auxlary problem 5 wth the functons p = µ β L 2 and q = β L 2 G γ L γ not dependng of tme, s an upper bound of the soluton of Eq. 3. Snce the upper bound s globally defned, we conclude that the soluton of Eq. 3 exsts globally n C R + ; L + and the estmaton of the tme after whch the soluton vanshes n a non neglgble subdoman follows Case σ = and γ >. In ths secton we consder Eq. 3 wth σ = and γ >. Theorem 2.5. Let σ =, γ >, β satsfy Assumpton and G L +. Then, there exsts an unque soluton of Eq. 3 n C [, T [ ; L wth the lower bound of the exstence tme ln µ G γ L γ G γ L T γ µ β L 2 G γ. L γ Moreover the soluton s nonnegatve. Proof. Consder the followng auxlary problem φ = pφ γ qφ, wth φ = φ. 6 The functons p and q are gven postve and contnuous on R +. By the Pcard Lndelöf theorem there exts an unque local soluton of 6. Moreover, φ t s nonnegatve when φ s nonnegatve. Then we approxmate the soluton of Eq. 3 by the sequence startng wth g t, v = G v and g n+ t, v s the unque postve soluton of the auxlary problem 6 wth p = β g n, q = β gn γ and the ntal data φ = G v, for any fxed v. We show the convergence of the sequence n the neghborhood of the ntal data usng the classcal contracton arguments. We denote by g the soluton of the auxlary problem 6 wth the functons p = β L 2 and q = µ β L 2 G γ L γ not dependng of tme and the ntal data φ = G v, for any fxed v. Snce the functon x x γ s convex, by

12 NONLINEAR KINETIC MODEL FOR THE FORMATION OF SWARM Proposton 2, g Lγ s ncreasng. We have t g = β g g γ β g γ g β L 2 g γ µ g γl γ g β L 2 g γ µ G γl γ g. It follows that g s an upper bound of the soluton of Eq. 3. We estmate the soluton of Eq. 6 for p and q that do not depend on tme by settng ψ = φ γ. We obtan φ t = pφ γ + q q pφ γ e γ qt and the tme of blowup for any ntal condton such that φ > whch ends the proof. ln q pφ T = γ, γ q γ φ, q γ p gven by Note that assumng the ntal data close to an equlbrum s not enough to conclude the global exstence. In fact, even n the case β =, the lower bound of the exstence tme tends to nfnty ff the ntal data satsfes G γ L γ = G γ L wth G L = ρ =. We wll see that ths condton corresponds to the steady state of Eq. 3, see Proposton 3. The next step conssts n dentfyng the exstence tme of the soluton of Eq. 3 wth the parameters σ = and γ >. We ndcate that the mathematcal result on the global exstence or blowup n a fnte tme s an open problem. To gve a meanng of the model, the soluton has to be at least n L γ. One may prove the global exstence of a smlar problem wth an addtonal dffusve operator, modelng the ndvdual reflecton, see [3, 33, 4]. 3. Formal hydrodynamc lmts. In ths secton we are gong to dscuss a formal dervaton of hydrodynamc lmts of Eq. n the case of a constant attractveness coeffcent,.e. γ x, ρ = γ. Ths s a prelmnary step and the rgorous results are stll open problems, n partcular n the case of self organzed nteractons σ γ <. The motvaton of the hydrodynamc lmt s to dentfy partcular asymptotc solutons n the perspectve to be compared to numercal solutons, see Secton 4. Snce the nteractons conserve the mass accordngly to Theorem 2., t ρ + x ρu = wth ρ t, x u t, x = vf t, x, v dv. 7 In order to estmate the macroscopc velocty u at least for small ε, we expand the soluton of Eq. usng the Hlbert expanson f t, x, v = = ε f t, x, v and we dentfy the terms of the same order of ε. The frst step of the dervaton of the hydrodynamc lmt conssts n determnng the dstrbuton f, the so called local equlbrum, such that Q f =. Equaton corresponds to an nfnte set of local equlbra, whch may lead to dfferent hydrodynamc lmts.

13 2 PARISOT MARTIN AND LACHOWICZ MIROS LAW Proposton 3. Let γ and β satsfy Assumpton. The local equlbra of Eq. are constant functons on ther support,.e. a nonnegatve functon f L + satsfes Q f = f and only f there exsts a measurable subset W t, x such that { ρ W, f v W, f t, x, v = where W = dw., otherwse Proof. Let I a,b g be I a,b g = β v, w gwg a v b dw dv 8 W W wth W = Supp g. Usng the Cauchy Schwarz nequalty on the product measure yelds I, g = β g γ / 2 w g γ / 2 v β g γ / 2 w g γ / 2 v dw dv W W W W W W βg γ wg 2 γ v 2 dw dv W W βg γ wg 2 γ v dw dv = I γ,2 γ g, βg 2 γ w W gv γ dw dv for any nonnegatve functon g L +. Snce f γ Q f = on the support of f, we have I, f = I γ,2 γ f. It follows that the local equlbrum satsfes the equalty n the Cauchy Schwarz nequalty,.e. there exsts a constant C such that for almost any v, w 2 we have β v, w f γ 2 γ v f w = Cβ v, w f 2 γ v f γ w. We conclude that the condton for an equlbrum soluton of Eq. to be constant on ts support s requred. Fnally, we easy check that the condton s suffcent. From now on, we defne the relevant local equlbrum of Eq. by the steady state of the homogeneous case Eq. 3 whch s Lyapunov stable. Such a choce of a local equlbrum s motvated by consderng Eq. along the trajectory, see [2] t f # = ε Q# [f], where f # t, x, v = f t, x vt, v. 9 Then for the Hlbert expanson we obtan Q f + εf = O ε. If the equlbrum s not Lyapunov stable, the solutons can leave the vcnty of the equlbrum and the term f s not controllable wth respect to ε. In such a case, the Hlbert expanson cannot be justfed. 3.. Macroscopc evoluton for entropy dsspatve nteractons. Proposton 4. Let β satsfy Assumpton. In both entropy dsspaton cases σ γ >.e. σ =, γ > as well as σ =, γ <, the only steady state of Eq. 3 that s Lyapunov stable n L s the constant functon n the doman. 2

14 NONLINEAR KINETIC MODEL FOR THE FORMATION OF SWARM 3 Proof. By Proposton 3, the steady states of Eq. 3 are constant functons on ther supports. We wll show that f the support W of the steady state g s not the whole doman, then the steady state s not stable. We ntroduce the perturbaton g L such that g dv = ; the perturbed state g ζ = g + ζ g, wth ζ >, s nonnegatve for ζ small enough and ts support s strctly larger than W,.e. W = Supp g ζ \ W has a nonzero measure. Such a perturbaton could always be obtaned by settng { W, f v W, g v = W, otherwse. Then for any w W, we have g w =, g w > and t g ζ w = σ β g ζ g γζ β g γ ζ g ζ = σ β g ζ g γ ζ γ σ β g γ ζ gζ + o ζ. When γ <, the frst term of the RHS domnates for ζ small. Smlarly, when < γ, the second term of the RHS domnates for ζ small. It follows t g ζ w > and then we conclude that the steady state cannot be stable. On the other hand, the maxmum prncple ensures that the constant functon n the whole doman s stable. Then, the relevant local equlbrum s gven by f d t, x, v = ρt,x, the so called dffusve pcture see [2]. Introducng f d n Eq. 7 we get u = v dv + O ε. The average velocty u s characterzed by the set of possble veloctes. For physcal reasons, assume that the doman s symmetrc,.e. for any v, we have v. It leads u = O ε and then t ρ = O ε. From the Hlbert expanson we obtan D fd Q f = σ γ f γ d β v, w dwf β f = v x f d. where D fd Q f s the Fréchet dervatve wth respect to the equlbrum. Note that f dv = accordng to the Hlbert expanson. We denote by L the set of functon g L such that g dv = and L : L L g β v, w g v g w dw. Proposton 5. Let β satsfy Assumpton and G L. There exsts a unque soluton g L of Lg = G. g Lg dv s postve for any nonzero g. Proof. We prove usng the followng Cauchy Schwarz nequalty I, g = β v, w g v β v, w g w dw dv W W β v, w g v 2 dw dv = I 2, g. W W

15 4 PARISOT MARTIN AND LACHOWICZ MIROS LAW wth I a,b defned by Eq. 8. We conclude that g Lg dv s nonnegatve. In addton, t vanshes only wth the equalty case of the Cauchy Schwarz nequalty, correspondng to the constant functon almost everywhere. Snce the average of the soluton vanshes, t follows that the soluton of g Lg dv = vanshes almost everywhere. Then, we prove usng the Fredholm theory. Snce β L 2, we only have to check that the homogeneous equaton Lg = does not admt any non trval soluton. Multplyng the homogeneous equaton by g and ntegratng yelds g Lg dv =. Usng we conclude that the only soluton of the homogeneous equaton s the zero soluton. Accordngly to Eq., the macroscopc equaton for the entropy dsspaton nteractons.e. σ γ > at the second order s gven by the followng nonlnear parabolc equaton κ t ρ ε x ρ γ xρ = O ε 2 wth κ = v L v dv γ. γ By Proposton 5 the dffuson coeffcent κ s postve snce t can be wrtten as χ Lχ dv κ = γ γ wth χ = L v. In addton, the nonlnearty ρ γ of the dffuson coeffcent s classcal for the generalzed Carleman type knetc models, see [3]. The lmt case when ρ vanshes, the so called regme of very fast dffuson s well known and descrbed n [9, 34, 29]. Note that the nonlnearty of the dffuson coeffcent can be modfed snce the nteracton rate β can be a functon of ρ. The dffuson coeffcent κ, for the classcal smple domans wth β v, w =, s gven by S d = { v R d : v = } 3 γ γd κ γ S γ d B d = { v R d : v } d 2 γ 2 γd γ d + 2 S γ d where > s a gven maxmal speed, S d s the surface area of the unt sphere n dmenson d,.e. S d = 2π d 2 and Γ s the Euler Gamma functon. In some Γ d 2 partcular cases, for example = B 3, β v, w = and γ = 3, the dffuson coeffcents s not a functon of the maxmal speed. Then the hydrodynamc lmt s defned for unbounded domans Macroscopc evoluton for self-organzed nteractons. Proposton 6. Let β v, w =. In both self organzed cases σ γ <.e. σ =, γ < as well as σ = and γ >, there s no steady states of Eq. 3 that are stable n L., 2

16 NONLINEAR KINETIC MODEL FOR THE FORMATION OF SWARM 5 Proof. In both self organzed cases t s obvous that for any pont v such that γ γ G v < G L γ, g t, v tends to zero when t tends to nfnty. By Proposton 3 the steady states of Eq. 3 are constant functons g on ther supports W,.e. γ γ g v = g L γ. We consder g ζ the soluton of Eq. 3 wth the ntal data G ζ defned as a perturbaton of the steady state G ζ = g + ζ W+ W wth the two subsets W W and W + W such that W = W + and ζ small. γ γ For any v W we have G ζ < G ζ L γ then t follows that g ζ t, v tends to zero for any v W. We conclude that the steady state s not stable. By Proposton 6 the local equlbrum have to be search among dstrbutons or measures. We may expect the local equlbrum n the form of a Drac functon f a t, x, v = ρ t, x δ v u t, x, where u s the mean velocty, see [, 23, 7] the so called algned pcture. Ths knd of lmt can be establshed consderng a dscrete set of possble veloctes,.e. for a fnte subset of Z, see [2]. Note that n ths framework, the problem of global exstence dsappear snce every norm are equvalent, and the l norm s preserved. Consderng a contnuous set of possble veloctes t s clear that Eq. has no meanng n the framework of dstrbutons or measure. Therefore we cannot derve macroscopc lmt usng classcal tools. The dervaton of a macroscopc lmt for Eq., even n the formal way, s an open problem. In the followng, we formulate a conjecture for the macroscopc lmt whch seems to correspond to the lmt of Eq. n the case of self organzed nteractons when ε tends to zero. We dscuss the relevance of the conjecture n Secton By Proposton 6, there s no local equlbrum n L but stll the L norm s fnte. We assume that the local equlbrum s mono knetc see [7],.e. f t, x, v = ρ t, x δ v ut, x. The man problem s the defnton of the mean velocty u t, x. Unfortunately, we are not able to solve the problem n the general case. However, n the case where β v, w s constant, the mean velocty s the velocty v such that the ntal datum F x, v s maxmal, f t s unque. Let us explan the strategy n one dmensonal framework for smplcty. It s clear that the asymptotc soluton of the homogeneous cases s the mono knetc functon wth the mean velocty gven by the velocty v such that the ntal condton F x, v s maxmal, f t s unque. More precsely, for the ntal data F, we defne U x such that F x, U x = max F x, v. v Then the soluton ρ t, x can be defned usng the characterstc method before a shock,.e. as long as the trajectores do not ntersect. In partcular, t s possble to create vacuum,.e. ρ =, f the velocty at left s smaller than the velocty at rght. At each pont where N trajectores ntersect, the soluton at the ntersecton pont s gven by the soluton of the homogeneous equaton 3 wth an ntal data composed by the N Drac functons,.e. G v = N ρ δ v u. = Note that t s exactly the case where the nteracton operator Q does not have a sense snce the functon G s not n L γ. We may however assume that the

17 6 PARISOT MARTIN AND LACHOWICZ MIROS LAW asymptotc soluton s gven by the velocty characterzed by the largest populaton, when ths s unque. More precsely the conjectured soluton after a shock s gven by g v = j M N = ρ Card M δ v u j wth M = { N ρ = max j N ρ j To llustrated t, we construct the conjectured soluton n the case of ntal data that are pecewse constant n Secton In the general case,.e. β s not constant, we are not able to defne the macroscopc lmt snce we are not able the defne the ntal mean velocty or the velocty after a shock. }. 4. Numercal solutons. The present secton s devoted to the numercal solutons of Eq.. We are gong to solve ndependently the nteractons between ndvduals and the transport n space, based on the formulaton along the trajectory 9. Such a splttng strategy s classcal for the numercal solutons of the knetc equatons. We present the numercal results n the dmensonal case. We propose several schemes for the nteracton operator Secton 4.2. The objectve s to desgn a scheme that s able to recover each soluton of Eq. 3 and not to restrctve n the lmt ε goes to zero. All these schemes can be easly extended to the mult dmensonal case. In Secton 4.3 we propose a soluton of the transport operator n the dmensonal case. For the Cartesan grd n space and n velocty, the strategy does not ntroduce the numercal dffuson, whch s requred to descrbe the formaton of swarms. The approxmaton of the transport operator cannot be extended to the mult dmensonal case or to the general grds. However, several more sophstcated approaches to desgn an ant dsspatve schemes can be found n the lterature see e.g. []. 4.. Numercal ntegraton n velocty. We start by ntroducng a numercal ntegraton wth respect to varable v. We ntroduce a Cartesan grd of the velocty space that s symmetrc = [, ],.e. Nv = {v } Nv N v such that v = dv wth dv = N v. We propose to use Newton Cotes formulas, wth postve wegh ω j >, denoted, to approxmate the nonlocal operators. The Newton Cotes formula of degree reads φ = dv N v j= N v ω j φ j, wth ω j = { 2, f j = N v or j = N v, otherwse, wth φ = φ v. In addton, we ntroduce the notaton of the dscrete nteracton rate β j = β v, v j >. The nonlocal operator β v, w φ w dw for the velocty v = v s naturally approxmated by φ = β φ = dv N v j= N v ω j β j φ j. Proposton 7 Dscrete local equlbrum. The local equlbra of the dscretzaton of Q usng the Newton Cotes formulas are the sets of values [G] Nv N v constant on ther supports,.e. there exsts a set I { N v,..., N v } and ρ

18 NONLINEAR KINETIC MODEL FOR THE FORMATION OF SWARM 7 such that [G] = { ρ I f I otherwse. wth I = dv I w. Proof. The proof s smlar to the contnuous case Proposton Numercal approxmatons of the nteractons. In the present secton we analyze several numercal schemes for soluton of the space homogeneous case 3. We ndcate the dependence wth respect to the Knudsen number ε. Our man objectve s to provde a numercal scheme stable for long tme step n order to use t at the macroscopc regme,.e. when ε tends to Euler explct scheme. The smplest way to solve Eq. 3 s probably the Euler explct tme scheme,.e. [g e ] n+ = [g e ] n + σ dt [ge ] n ε [g e ] n γ [g e ] n γ [g e] n, 3 wth the tme step dt, [g e ] n s an approxmaton of the soluton g t n, v and the dscrete ntal data [g e ] = [G] = G v. Proposton 8 Propertes of the Euler explct scheme 3. Let the ntal data be nonnegatve and ntegrable,.e. [G] and [G] <. The scheme [g e ] 3 satsfes the followng propertes: The sequence [g e ] s an approxmaton of the soluton of Eq. 3 n order n tme. The approxmaton [g e ] satsfes the mass conservaton,.e. [g e ] n,r = [G]. Under the CFL condton Ce pos dt ε wth { Ce pos = max σ N v N v [ge ] n γ [g e] n [g e ] n γ }, the approxmaton [g e ] s non negatve,.e. [g e ] n. v For the soltarous cases,.e. σ γ >, and under the CFL condton Ce max dt ε wth N v γ [g e ] n Ce max = dv max ω j β j γ [g e ] n j [g e ] n j [ge ] n N v N v [g e ] n [g e] n j, j= N v [g e] n j [ge]n the approxmaton [g e ] satsfes the maxmum prncple mn [G] N j [g e ] n max [G] v N v N j. v N v Proof. The propertes, and are obvous. In order to prove v we assume that the approxmaton n teraton n s nonnegatve whch s true at the ntal data [G]. The scheme [g e ] 3 can be wrtten n the form [g e ] n+ = dt dv + σ ε σ dt dv ε N v j= N v [g e] n j [ge]n N v j= N v [g e] n j [ge]n [g e ] n γ [g e ] n j ω j β j [g e ] n [g e] n j [g e ] n γ ω j β j [g e ] n [g e] n j γ γ [g e ] n j [g e ] n [g e] n j [g e] n [g e ] n [g e] n j [g e ] n j.

19 8 PARISOT MARTIN AND LACHOWICZ MIROS LAW For the soltarous cases,.e. σ γ >, for any a, b R +, the functon σ aγ b γ a b ab s nonnegatve. Then, under the CFL condton v, the approxmaton n tme teraton n + s a convex functon of the approxmaton at tme n. We conclude the maxmum prncple of the dscrete approxmaton Euler sem mplct scheme. The Euler explct scheme 3 s stable under the CFL condton, see Proposton 8. The CFL condtons could become restrctve, n partcular consderng macroscopc regme,.e. when ε tends to zero. To avod ths dffculty, t s classcal to use Euler mplct scheme. The nonlnear term requres usng of teratve process based on the lnearzaton. For smplcty, we look for the numercal strategy that does not requre the soluton of matrx systems. However, t s not possble to fnd a sem mplct scheme satsfyng all the propertes of the contnuous soluton, n partcular the dscrete verson of the mass conservaton Theorem 2.. Ths conservaton s the key pont of stablty of the numercal scheme snce for the self organzed models,.e. σ γ <, t s the only norm decreasng n tme. In order to fx the mass conservaton at the dscrete level, we propose a correcton step usng an operator P ρ : R 2Nv+ + \ {} R 2Nv+. Lemma 4.. Assume that the operator P ρ satsfes the followng propertes: { } The mage of the operator P ρ s {φ j } R 2Nv+ + φ = ρ. For the gregarous cases,.e. σ γ <, the operator P ρ s nonnegatve. Moreover, for any N v N v such that φ =, we have P ρ =. For the soltarous cases,.e. σ γ >, any nterval of R + s stable by the { operator} P ρ,.e. for any a b and {φ } [a, b] 2Nv+, we have P ρ {φ j } j [a, b] 2Nv+. v For any {φ } R 2Nv+ \ {} such that φ = ρ and the perturbaton {ψ } R 2Nv+ such that ψ = O ν and {φ + ψ } R 2Nv+ \ {}, we have {φ j + ψ j } j = φ + O ν. P ρ In the present paper, we set P ρ {φ j } j = f φ = and P ρ ρ ρ {φ j } j = φ + φ ρ ρ ρ 2 φ φ 2 + φ 2 φ + φ 2 ρ ρ max φ 2 φ, φ 2 + φ 4 otherwse. The proofs of the propertes of Lemma 4. usng the formula 4 are obvous. Note that other defntons of the operator P ρ are possble. We desgn the sem mplct scheme [g ] n,r+ [g ] n,r = ε [g ] n, where [g ] n+, [g ] n, ε + dt { } = P [G] [g ] n,r j, where j + dt σ + [g ] n,r [g ] n,r γ + σ [g ] n,r γ [g ] n,r σ + [g ] n,r γ + σ [g ] n,r [g ] n,r γ, 5 = lm r [g ] n,r and [g ], = [G] = G v. The dscrete unknown s an approxmaton of the soluton of Eq. 3 at the pont t n, v. If the approxmaton at the prevous tme step vanshes,.e. [g ] n,r =, we have [g ] n,r = except n the case of negatve gregarous nteractons,.e. σ = and γ <, where [g ] n,r s not defned. By the contnuous analyss, see Theorem 2.4, we known

20 NONLINEAR KINETIC MODEL FOR THE FORMATION OF SWARM 9 that the only nonnegatve soluton n ths case s the vanshng soluton. Then we can extent the numercal scheme settng [g ] n,r = f [g ] n,r = for any σ and γ. Proposton 9 Propertes of the Euler sem mplct scheme 5. Let the ntal data be ntegrable,.e. [G] < and nonnegatve,.e. [G]. The scheme [g ] 5 satsfes the followng propertes: The sequence [g ] s an approxmaton of the soluton of Eq. 3 n order n tme. The approxmaton [g ] satsfes the mass conservaton,.e. [g ] n,r = [G]. The approxmaton [g ] s nonnegatve,.e. [g ] n,r. Moreover, for any N v N v such that the ntal data [G] are postve, the approxmaton [g ] n,r s postve. v For the soltarous case,.e. σ γ >, the approxmaton [g ] satsfes the maxmum prncple,.e. mn [G] N j [g ] n,r max [G] v N v N j. v N v Proof. It s clear that [g ] n,r dt s an approxmaton of order n tme of the soluton of Eq. 3. Then usng Lemma 4..v, we conclude the property. The property s a corollary of the Lemma 4... Usng the recursve argument, the non negatvty postvty s obvous snce all the term are nonnegatve postve. Assume that the prevous approxmaton [g ] n,r satsfes the maxmum prncple mn [g ] n, N j = m n [g ] n,r M n = max [g ] n, v j N v N j. v j N v Note that the ntal approxmaton [g ] n, satsfes ths assumpton. We show that [g ] n,r satsfes the property n the case of postve nteracton,.e. σ =. For any X R we have [g ] n, X + dt [g ] n,r [g ] n,r ε [g ] n,r γ [g ] n,r γ X X = ε + dt [g ] n,r γ. The frst term of the nomnator s clearly nonpostve wth X = M n and nonnegatve wth X = m n. Snce γ the functons x γ s decreasng and x γ s ncreasng then [g ] n,r [g ] n,r γ [g ] n,r γ M n [g ] n,r [g ] n,r γ Mn γ. The smlar result holds for the lower bound. We conclude m n [g ] n,r M n. We proceed smlarly for the negatve nteracton case. More precsely, we wrte ε [g ] n, [g [g ] n,r X + dt ] n,r γ [g ] n,r [g ] n,r [g ] n,r γ X X = ε + dt [g ] n,r [g ] n,r γ. Snce γ, the functon x γ s ncreasng and [g ] n,r γ [g ] n,r [g ] n,r [g ] n,r γ M n [g ] n,r Mn γ [g ] n,r [g ] n,r γ M n The smlar result holds for the lower bound. Fnally we conclude the property v usng Lemma 4...

21 2 PARISOT MARTIN AND LACHOWICZ MIROS LAW Note that the sem mplct scheme 5 can be used n the lmt ε =. Thanks to the teratve process, when r tends to nfnty, the soluton wth ε = tends to the asymptotc soluton of Eq. 3 whch s gven by the steady state Proposton 7. In practce, we perform several tests postve or negatve nteractons, soltarous or gregarous behavor, convex or concave nteracton rate β and the teratve process converges n any cases usng the sem mplct scheme Non lnear scheme. The explct and mplct Euler schemes are based on a lnearzaton of the nteracton operator. Then the prevous schemes 3, 5 conserve the vanshng ponts,.e. for any ponts v such that G v =, we have [g e ] n = [g ] n, =. Even f ths soluton s relevant from mathematcal pont of vew, t s not satsfactory at the bologcal level, see Remark. In addton, none of the prevous schemes are able to vansh n a fnte tme n the case of negatve gregarous nteracton, see Theorem 2.4. The objectve of ths secton s to desgn a numercal scheme able to recover the soluton n the case of no Lpschtz nteracton coeffcent γ <,.e. the soluton ables to leave zero n case of postve soltarous nteracton and the soluton ables to vansh n a fnte tme n case of negatve gregarous nteracton. As we mentoned, ths soluton requred the non lnear structure of the equaton. We propose to base the numercal scheme on the followng nonlnear ODE problem εσφ = pφ γ qφ wth φ = φ. 6 Assumng γ, we use the substtuton of unknown ψ = φ γ to get the lnear ODE problem εσψ = γ p qψ wth ψ = ψ. 7 Then t s possble to use the explct or mplct Euler schemes to get an approxmaton of soluton of Eq. 7. However, ths schemes lead to restrctve condtons on the tme step n the case of gregarous nteracton σ γ. In fact, the tme step ensurng the postvty of the numercal soluton vanshes when the soluton vanshes n some ponts. To overcome ths drawback, we use the exact soluton of the ODE 7 assumng that the parameters p and q are two gven constants. From now on, we denote by ψ t, ψ, p, q the soluton of the lnear ODE 7 and we have the followng explct formula ψ t, ψ, p, q = p q + ψ p exp σ γ q t. 8 q ε For the nonnegatve ntal data ψ and postve parameters p and q, the soluton Eq. 8 s nonnegatve n the soltarous case,.e. σ γ >. For the negatve gregarous case,.e. σ = and γ <, the only nonnegatve soluton of the nonlnear ODE 6 s gven by the extenson of 8 by zero, see Secton For the postve gregarous case,.e. σ = and < γ, the soluton of the nonlnear ODE vanshes only f φ =. In addton, the soluton of the nonlnear ODE 6 blows up, see Fnally, we desgn the nonlnear scheme [g n ] n,r+ { } = P [G] [g n ] n,r j, where j γ [g n ] n,r = max, ψ dt, [g n ] n,, [gn ] n,r, [g n ] n,r γ γ, 9 where [g n ] n+, = lm r [g n ] n,r and [g n ], = [G] = G v.

22 NONLINEAR KINETIC MODEL FOR THE FORMATION OF SWARM 2 For the postve gregarous case,.e. σ = and < γ, the numercal scheme 9 s defned only f, for any N v N v such that [g n ] n, >, the tme step satsfes [g ε ln n] n,r γ [g n] n,r [g n] n, γ dt γ [g n ] n,r γ. 2 Ths tme step corresponds to the tme of blow up of nonlnear ODE 6. Proposton Propertes of the nonlnear scheme 9. Let the ntal data be ntegrable and nonnegatve: [G] < and [G]. The scheme [g n ] 9, under the CFL condton 2, satsfes the followng propertes: The sequence [g n ] s an approxmaton of the soluton of Eq. 3 n order n tme. The approxmaton [g n ] satsfes the mass conservaton [g n ] n,r = [G]. The approxmaton [g n ] s nonnegatve [g n ] n,r. v For the soltarous case,.e. σ γ >, the approxmaton [g n ] n, satsfes the maxmum prncple,.e. mn [G] N j [g n ] n,r max [G] v N v N j. v N v Proof. It s clear that [g n ] n,r s an approxmaton n order n tme of the soluton of Eq. 3. Then usng Lemma 4..v we obtan. The property s a corollary of the Lemma 4... The property s obvous by the usng the formula 9 and Lemma 4... For the soltarous cases,.e. σ γ >, for a fxed value of {[g n ] n,r }, the new approxmaton [g n ] n,r s a monotonous functon of the tme step t. More precsely, the new approxmaton [g n ] n,r s bounded by prevous step approxmaton [g n ] n, [gn] and tends to n,r γ [g n] n,r γ when t tends to nfnty. In order to prove the maxmum prncple, we have to show that the two bounds are larger than the mnmum m n = mn Nv j N v [g n ] n, max Nv j N v [g n ] n, j maxmum prncple j and smaller than the maxmum M n =. Assume that the prevous approxmaton [g n ] n,r satsfes the mn [g n ] n, N j = m n [g n ] n,r M n = max [g n ] n, v j N v N j. v j N v Note that the ntal approxmaton [g n ] n, satsfes ths assumpton. Then for γ <, we have the followng nequaltes [gn ] n,r γ m γ n [g n ] n,r = [g n ] n,r γ [g n ] n,r γ [g n ] n,r γ M n γ. Smlarly, for < γ, we have the followng nequaltes [g n ] n,r m γ n [g n ] n,r γ = [g n ] n,r [g n ] n,r γ [g n ] n,r Mn γ. Usng the recursve argument and Lemma 4.., we obtan v Spacal evoluton. In the followng, we propose a numercal strategy for the space dependng PDE n the whole space of dmenson. The boundary condtons are not consdered n ths work and the numercal smulaton are performed usng the perodc doman. We propose the Euleran dscretzaton on the Cartesan grd x k = k dx, wth dx the space step. The choce of the Euleran dscretzaton

23 22 PARISOT MARTIN AND LACHOWICZ MIROS LAW s motvated by the dffuson pcture n the case of soltarous behavor, see Secton 3.. The dffuson equaton are classcally solved usng Euleran dscretzaton. In addton, we set t the tme step for the transport scheme. The queston of the numercal soluton n the mult dmenson case and wth the general grd wll be addressed n further works Up wnd scheme. Once the velocty doman s dscretzed on a grds, t s classcal to use an explct up wnd scheme to treat the transport wth constant velocty. We defne by [fz u ] n k, the approxmaton of soluton f of Eq. at pont t n, x k, v usng the up wnd scheme and one of the nteracton schemes defned n Secton 4.2,.e. z {e,, n}. We decompose the up wnd scheme n the four steps:. We construct an approxmaton of the dstrbuton at the nterfaces x k + dx 2 usng the dstrbuton at the left for the postve velocty and the velocty at the rght for the negatve velocty. More precsely we set [fz u ] n [fz u ] n k+, f < k+ / = [f u z ]n k, +[f u z ]n k+, 2, 2 f = [fz u ] n k, f <. 2. We estmate the dstrbuton at the nterface after nteracton,.e. [fz u ] n t } k+ / = Q 2, z, {[f 2, z u ] n k+ / 2,j. j wth the nteracton step Q z, T, {φ j } j that s the soluton of the scheme z {e,, n} proposed n Secton 4.2 at the fnal tme T usng the ntal data [G] = φ. 3. We transport the dstrbuton at the nterface usng a classcal up wnd scheme for the constant velocty. More precsely, we set [f u z ] n+ k, = v t dx [f u z ] n k, + t dx v + [f u z ] n k / 2, + v [f u z ] n k+ / 2,,. 2 It s well known that the up wnd scheme s stable under the CFL condton v t dx. 4. We estmate the dstrbuton at the grd pont after nteracton,.e. [f u z ] n+ k, = Q z, t 2, { } [fz u ] n+ Unfortunately we wll see from the smulaton n Secton that the numercal dffuson s too large to represent the swarms. In fact the numercal dffuson s proportonal to the CFL parameter v t dx. In the followng Secton 4.3.2, we propose a numercal scheme based on a tme splttng accordngly to the velocty, such that the CFL parameter lnks to the consdered velocty s set to and the numercal dffuson vanshes Ant dffusve scheme. We defne by [fz a ] n k, the approxmaton of f soluton of Eq. at pont t n, x k, v usng the ant dffusve scheme and one of the nteracton schemes defned n Secton 4.2,.e. z {e,, n}. The strategy s based on the fact that when v t dx =, the trajectores ntersect the grd ponts and the numercal dffuson vanshes. To set the CFL parameter to for any velocty, we use dfferent k,j j.

24 NONLINEAR KINETIC MODEL FOR THE FORMATION OF SWARM 23 tme step accordngly to the velocty. In addton, to treat the nteracton between the drect neghbors, we have to consderate the nteracton durng the transport step. Let us decompose the ant dffusve scheme n fve steps. The steps, 2, and 4 are smlar to those n the strategy proposed n the prevous Secton The transport step 3 s smlar as n the case of a transport wth the CFL parameter set to. In addton, a specal treatment of the part of the populaton whch does not move,.e. =, s requred. Fnally, the step 5 s an actualzaton of the soluton when the dfferent tme step are synchronzed. Note that ths synchronzaton s realzable by the fact that the Cartesan grd s used. The generalzaton to a general grd requred more sophstcated treatment, see [].. We construct an approxmaton of the dstrbuton at the nterfaces x k + dx 2 usng the dstrbuton at the left for the postve velocty and at the rght for the negatve velocty,.e. we set [fz a ] n,l k+, f < [fz a ] n,l k+ / = [f a z ]n,l k, +[f a z ]n,l k+, 2, 2 f = [fz a ] n,l k, f <. 2. We estmate the dstrbuton at the nterface after nteracton,.e. [fz a ] n,l dx { } k+ / = Q 2, z, 2 dv, [fz a ] n,l k+ / 2,j. j 3. We transport the dstrbuton at the nterface consderng that only the fast enough velocty,.e. v > l dv, passes through the nterface,.e. we set [fz a ] n,l k+ / 2, f l N v [fz a ] n,l k / 2, f l N v < < [fz a ] n,l+ k, = f = 22 2 [f a z ]n,l k, [f a z ]n,l k / 2, [f a z ]n,l k / 2, + [f a z ]n,l k, [f a z ]n,l k+ / 2, [f a z ]n,l k+ / 2, [fz a ] n,l k+ / 2, f < < N v l [fz a ] n,l k / 2, f N v l. Because of the reconstructon at the nterface, the dstrbuton s splt and leads to numercal dffuson only for = wthout ths specal treatment. To overcome ths dffculty we reconstruct the approxmaton at the grd pont x k consderng the rato of the dstrbuton at the nterface comng from the pont x k. 4. We estmate the dstrbuton at the grd pont after nteracton,.e. [f a z ] n,l+ k, = Q z, dx 2 dv, { } [fz a ] n,l+. 5. We terate the steps to 4 untl all the veloctes pass trough the nterface,.e. for l N v and we set the approxmaton at the new tme step [fz a ] n+, k, = [fz a ] n,nv k,. The numercal scheme can be nterpreted as a tme splttng wth N v loops wth the sub tme step dx, wth the larger speed. In each loop, we have splt the transport step and 3 and the nteracton step 2 and 4 nto 2 steps. It follows that the new approxmaton [fz a ] n+, k, s an approxmaton of the soluton of Eq. k,j j

25 24 PARISOT MARTIN AND LACHOWICZ MIROS LAW at the pont t n + dx dv, x k, v. In addton, the numercal unknown [f a z ] n,l k, can be consdered as an approxmaton n the pont t n + l dx, x k, v. 5. Numercal smulatons and valdaton. 5.. Numercal smulatons of the space homogeneous case. In the present secton we compare the numercal schemes of the nteracton operator descrbed n Secton 4.2. We consder n the space homogeneous case and wth = [, ], =.9, dv = The ntal data are defned by G v = max,.75 sn 4. + v 2 π e v.752 2, 23 the nteracton rate s β v, w = e v w2 and the Knudsen number s ε = tme [g e ] tme [g ] velocty tme [g n ] densty densty densty Fgure 2. Numercal result of the negatve soltarous nteracton case,.e. σ =, γ = 2. Explct scheme 3 [g e ] frst lne, semmplct scheme 5 [g ] second lne, and non-lnear scheme 9 [g n ] thrd lne. In Fgure 2 we represent the numercal solutons n the case of negatve soltarous nteractons,.e. σ =, γ = 2. The results of the proposed schemes 3, 5 and 9 are almost the same. The vanshng areas are preserved along the smulaton and the steady state s the constant soluton on the support of the ntal data. Note that n ths case, the soluton s unque. In Fgure 3 we present the numercal solutons n the case of postve soltarous nteractons,.e. σ =, γ =.5. The results of the lnear schemes 3 and 5 are almost the same. The vanshng areas are preserved along the smulaton and the steady state s the constant soluton on the support of the ntal data. The

26 NONLINEAR KINETIC MODEL FOR THE FORMATION OF SWARM tme [g e ] tme [g ] velocty tme [g n ] densty densty densty Fgure 3. Numercal result of the postve soltarous nteracton case,.e. σ =, γ =.5. Explct scheme 3 [g e ] frst lne, sem mplct scheme 5 [g ] second lne, and nonlnear scheme 9 [g n ] thrd lne. numercal solutons usng the nonlnear scheme 9 lead to the constant steady state on the whole doman. Both solutons, constant on the ntal data support as well as constant on the whole doman, are steady states of Eq., but only the latter s Lyapunov stable see Proposton 4. It follows that only soluton satsfyng Remark s the soluton approached by the scheme 9. We wll see that ths property leads to mportant consequences at the macroscopc level see Secton In Fgure 4, we represent the numercal solutons n the case of negatve gregarous nteractons,.e. σ =, γ =.5. The CFL condton of the explct scheme 3 Proposton 8 s too restrctve n the velocty pont grd such that the soluton s small. In fact the tme step satsfyng the CFL condton tends to vansh wth the soluton. The results wth the two other schemes 5 and 9 are almost the same. Note that the asymptotc soluton s a Drac functon. However, the support of the Drac functon v =.2 s not the velocty for whch the ntal datum s maxmal v = 7.2. The correspondence between the support of the asymptotc soluton and the localzaton of the maxmum of the ntal datum s only vald when the nteracton rate s constant,.e. β = whch s not the case n the present smulaton. In Fgure 5 we represent the numercal solutons n the case of postve gregarous nteractons,.e. σ =, γ = 2. The results of the two lnear schemes 3 and 5 are almost the same. We can clearly dentfy a tme t 5.2 after whch the

27 26 PARISOT MARTIN AND LACHOWICZ MIROS LAW tme [g ] velocty tme [g n ] densty densty Fgure 4. Numercal result of the negatve gregarous nteracton case,.e. σ =, γ =.5. Sem-mplct scheme 5 [g ] frst lne, and non-lnear scheme 9 [g n ] second lne tme [g e ] tme [g ] velocty tme [g n ] densty densty densty Fgure 5. Numercal result of the postve gregarous nteracton case,.e. σ =, γ = 2. Explct scheme 3 [g e ] frst lne, semmplct scheme 5 [g ] second lne, and non-lnear scheme 9 [g n ] thrd lne. soluton s concentrated n a sngle velocty grd pont. In addton, the teratve process of the nonlnear scheme 9 does note converge at ths tme. From now on we refer to ths tme as the tme of numercal blow up. Usng a smaller velocty step, the tme of numercal blow up seams to be unchanged. Ths observatons are

28 NONLINEAR KINETIC MODEL FOR THE FORMATION OF SWARM 27 non mathematcal arguments showng a probable blow up of the soluton n the postve gregarous case Numercal smulatons of the hydrodynamc regmes. The followng secton s devoted to the numercal valdaton of the asymptotc regmes dentfed n Secton 3. We compare the dfferent schemes proposed n Secton 4.2 and Secton 4.3 and dscuss the relevance of the soluton. Due to the CFL condton functon of the Knudsen number ε, we do not consder the explct scheme for nteracton 3. We consder the mono knetc ntal data F x, v = R x δ v Ux defned by ρ =. f x =. x x 2 =.35 u =.8 f x x x 2 ρ 2 =. f x 3 =.4 x x 4 =.5 ρ 3 =.2 f x 4 x x 5 =.6 u 2 =.2 f x 3 x x 4 R x = and U x = u 3 =. f x 4 x x 5 ρ 4 = 22.2 f x 6 =.7 x x 7 =.8 u 4 =.3 f x 6 x x 7 ρ 5 =.3 f x 7 x x 8 =.9 u 5 =. f x 7 x x 8 elsewhere 24 n a [, ]-perodc doman The dffusve pcture. In the followng secton, we consder the postve soltarous case,.e. σ = and γ =.5 wth β = of the ntal data 24. In the present secton, we assume that the agent can move untl a maxmum velocty set to =.,.e. = B.. To get a relevant numercal soluton of the nteracton operator, the dscrete space step dx should be smaller than the mean fee path l. Thus the Knudsen number has to be larger than the dmensonless space step,.e. ε = l L Snce the soluton s perodc, the characterstc length scale s L =. All the numercal results gven n ths secton are obtaned usng the space step dx = 3 and the fxed Knudsen number ε = dx L = 3. In Fgure 6, we present the numercal solutons obtaned wth the dfferent dx L. schemes proposed above and the ntal datum 24. The space x s plotted on the abscssa and the tme t s plotted on the ordnate. Accordngly to Secton 4 the macroscopc regme s gven by the nonlnear parabolc equaton. In the frst lne of Fgure 6 we present the numercal soluton obtaned wth the nonlnear parabolc equaton neglectng the term n ε 2 and usng the classcal centered mplct scheme. Usng the estmaton realzed n 2, the dffuson coeffcent of the asymptotc nonlnear parabolc equaton s set to κ = 3 / 2 3π. In the second lne of Fgure 6, we present the numercal soluton obtaned wth the up wnd scheme n space 2 and the sem mplct scheme for the nteractons 5 [f u ]. The approxmated soluton s very far from the macroscopc soluton,.e. frst lne of Fgure 6. As we have seen n Secton 5. the local equlbrum of the scheme 5 s not the constant soluton n the velocty doman. It does not converge to the Lyapunov stable equlbrum so called dffusve pcture used to estmate the nonlnear parabolc equaton. More precsely, the set of velocty of the agent s stable usng the sem mplct scheme 5, thus the soluton vanshes n each velocty grd ponts on whch the ntal condton vanshes n the whole space doman and at any tme,.e. f x R F x, v = then t R + and x R f t, x, v =. Ths drawback clearly lnks to the use of the sem mplct scheme 5 and we obtan smlar result usng the sem mplct scheme 5 wth the ant dffusve scheme n space 22 n the thrd lne of Fgure 6. The velocty of the agent s

29 28 PARISOT MARTIN AND LACHOWICZ MIROS LAW tme macroscopc eq. tme [f u ] tme [f a ] tme [f u n ] tme [f a n ] poston densty densty densty densty densty Fgure 6. Numercal result of the dffusve pcture,.e. σ =, γ =.5. Nonlnear parabolc equaton frst lne, antdffusve space scheme 22 wth nonlnear scheme for the nteractons 9 [fn] a second lne, upwnd space scheme 22 wth nonlnear scheme for the nteractons 9 [fn u ] thrd lne, ant-dffusve space scheme 22 wth sem mplct scheme for the nteractons 5[f a ] fourth lne, upwnd space scheme 22 wth sem mplct scheme for the nteractons 5[f u ] ffth lne.

30 NONLINEAR KINETIC MODEL FOR THE FORMATION OF SWARM 29 even more clear snce the numercal dffuson s lmted. In the case of postve soltarous nteractons we dscredt ths soluton consderng bologcal arguments, see Remark. However, n the case of negatve soltarous nteractons, the soluton s unque, see Theorem 2.2, and the scheme of the nteracton operator leads to the unstable equlbrum. The hydrodynamc lmt s thus not defned n the negatve soltarous nteractons case, except when the ntal datum s separated from zero. In the fourth and ffth lnes of Fgure 6 we present the numercal solutons obtaned wth the nonlnear scheme for the nteractons 9. More precsely, the fourth lne of Fgure 6 s obtaned usng the up wnd scheme n space 2 [fn u ] and the ffth lne of Fgure 6 s obtaned usng the ant dffusve n space 22 [fn]. a The results are qualtatvely smlar to the macroscopc soluton,.e. frst lne of Fgure 6. However, the soluton s sgnfcantly more dffusve than the macroscopc soluton n partcular usng the up wnd scheme [fn u ]. In addton, the dffuson n the areas wth large densty s clearly faster, whch characterze that the nonlnearty of the dffuson s not well-represented. In the case of the negatve soltarous nteractons,.e. σ = and γ = 2, the nonlnear scheme for nteracton 9 preserves the ponts n the velocty grd such that the dstrbuton vanshes, as well as the sem mplct scheme 5. The soluton obtaned n ths case s qualtatvely the same as presented n the second lne of Fgure 6 for the up wnd scheme n space 2 or the thrd lne of Fgure 6 for the ant dffusve scheme n space The algned pcture. In the followng secton we consder the negatve gregarous case,.e. σ = and γ =.5, wth β = n the macroscopc lmt ε =, for the ntal datum 24. The followng results are smlar to the postve gregarous nteractons case,.e. σ = and γ = 2, n the macroscopc lmt ε =, except that the non lnear fxed pont for nteractons converges slowly. Frst we estmate the conjectured soluton followng the rules desgn n Secton 3.2 for an ntal condton that s pecewse constant. At the ntal tme, two dstrbutons meet at pont x 4. Snce u 2 > u 3, the trajectory startng from x 4 s a shock. Snce ρ 2 < ρ 3, the shock s movng wth the velocty u 3 and the densty from the left part gathers on ths trajectory: we have a space Drac functon S 4 δ x x 4 + u 3 t wth the mass S 4 = u 2 u 3 tρ 2 for a tme t small enough to do not ntersect another trajectory. Smlarly, at the ntal tme, two dstrbutons meet at pont x 7. However, snce u 4 < u 5, there s no nteractons and the two dstrbutons have ther veloctes, creatng an empty area between them. We proceed smlarly untl all the populatons tend to the same drecton. In Fgure 7 we present the conjectured soluton for the ntal datum 24 wth x n abscssa and t n ordnate. t ρ = ρ = t x ρ x 2 x 3 ρ 2 x 4 ρ 3 x 5 x 6 ρ 4 x 7 ρ 5 x 8 x Fgure 7. Conjecture of the hydrodynamc gregarous soluton wth ntal data that s pecewse constant gven by 24.

31 3 PARISOT MARTIN AND LACHOWICZ MIROS LAW tme [f u ] tme [f a ] dx=-3 ] dx >-3 tme [f a poston densty densty densty Fgure 8. Numercal result of the algned pcture,.e. σ =, γ =.5 and usng the sem mplct nteracton scheme 9. Upwnd space scheme 2 [f u ] frst lne, ant-dffusve space scheme 22 [f u] wth a space step dx = 3 second lne, and antdffusve space scheme 22 [f u ] wth a space step dx = 999 thrd lne. In Fgure 8 we present the numercal solutons obtaned wth the dfferent schemes proposed above and the ntal datum 24. The space x s plotted on the abscssa and the tme t s plotted on the ordnate. We use the sem mplct scheme 5 to treat the nteracton operator. The nonlnear nteracton scheme 9 cannot be used n the asymptotc lmt ε = n the case of gregarous nteracton σ γ <. However, smlar results were obtaned for the nonlnear nteracton scheme 9 wth the small Knudsen numbers,.e. ε < 3. The velocty step set to dv =.. The results wth the other velocty steps are exactly the same as long as the ntal veloctes are represented n the velocty grd. In the frst lne of Fgure 8, we use the up wnd scheme n space 2 [f u ] wth a space step dx = 3. The numercal dffuson makes the numercal approxmaton useless because the part of the dstrbuton able to pass through an nterface s proportonal to the tme step. Snce the tme step s small for the stablty reasons, the dstrbuton, ntally n a mesh pont, s generally larger than the part ncomng from the neghbors. Thus after applcaton of the nteracton operator, the populaton tends to the drecton of the ntal data n each pont, except n the part where