{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T19:48:09Z","timestamp":1776714489935,"version":"3.51.2"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,2,9]],"date-time":"2025-02-09T00:00:00Z","timestamp":1739059200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,2,9]],"date-time":"2025-02-09T00:00:00Z","timestamp":1739059200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100006256","name":"Universit\u00e0 degli Studi dell\u2019Aquila","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100006256","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2025,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We investigate various versions of multi-dimensional systems involving many species, modeling aggregation phenomena through nonlocal interaction terms. We establish a rigorous connection between kinetic and macroscopic descriptions by considering the small inertia limit at the kinetic level. The results are proved either under smoothness assumptions on all interaction kernels or under singular assumptions for <jats:italic>self-interaction<\/jats:italic> potentials. Utilizing different techniques in the two cases, we demonstrate the existence of a solution to the kinetic system, provide uniform estimates with respect to the inertia parameter, and show convergence toward the corresponding macroscopic system as the inertia approaches zero.\n<\/jats:p>","DOI":"10.1007\/s00332-025-10134-x","type":"journal-article","created":{"date-parts":[[2025,2,9]],"date-time":"2025-02-09T10:25:56Z","timestamp":1739096756000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Small Inertia Limit for Coupled Kinetic Swarming Models"],"prefix":"10.1007","volume":"35","author":[{"given":"Young-Pil","family":"Choi","sequence":"first","affiliation":[]},{"given":"Simone","family":"Fagioli","sequence":"additional","affiliation":[]},{"given":"Valeria","family":"Iorio","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,2,9]]},"reference":[{"key":"10134_CR1","first-page":"1","volume-title":"Lecture Notes on Optimal Transport Problems","author":"L Ambrosio","year":"2003","unstructured":"Ambrosio, L.: Lecture Notes on Optimal Transport Problems, pp. 1\u201352. Springer, Berlin (2003)"},{"key":"10134_CR2","series-title":"Lectures in Mathematics","volume-title":"Gradient Flows in Metric Spaces and in the Space of Probability Measures","author":"L Ambrosio","year":"2008","unstructured":"Ambrosio, L., Gigli, N., Savar\u00e9, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics, 2nd edn. Birkh\u00e4user Verlag, Basel (2008)","edition":"2"},{"issue":"5","key":"10134_CR3","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1007\/s11401-009-0191-5","volume":"30","author":"AL Bertozzi","year":"2009","unstructured":"Bertozzi, A.L., Laurent, T.: The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels. Chin. Ann. Math. Ser. B 30(5), 463\u2013482 (2009)","journal-title":"Chin. Ann. Math. Ser. B"},{"issue":"1","key":"10134_CR4","doi-asserted-by":"crossref","first-page":"45","DOI":"10.4310\/CMS.2010.v8.n1.a4","volume":"8","author":"AL Bertozzi","year":"2010","unstructured":"Bertozzi, A.L., Brandman, J.: Finite-time blow-up of $$L^\\infty $$-weak solutions of an aggregation equation. Commun. Math. Sci. 8(1), 45\u201365 (2010)","journal-title":"Commun. Math. Sci."},{"issue":"3","key":"10134_CR5","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1088\/0951-7715\/22\/3\/009","volume":"22","author":"AL Bertozzi","year":"2009","unstructured":"Bertozzi, A.L., Carrillo, J.A., Laurent, T.: Blow-up in multidimensional aggregation equations with mildly singular interaction kernels. Nonlinearity 22(3), 683\u2013710 (2009)","journal-title":"Nonlinearity"},{"issue":"1","key":"10134_CR6","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1002\/cpa.20334","volume":"64","author":"AL Bertozzi","year":"2011","unstructured":"Bertozzi, A.L., Laurent, T., Rosado, J.: $$L^p$$ theory for the multidimensional aggregation equation. Comm. Pure Appl. Math. 64(1), 45\u201383 (2011)","journal-title":"Comm. Pure Appl. Math."},{"key":"10134_CR7","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1002\/mma.638","volume":"28","author":"M Bodnar","year":"2005","unstructured":"Bodnar, M., Velazquez, J.-J.: Derivation of macroscopic equations for individual cell-based models. A formal approach. Math. Methods Appl. Sci. 28, 10 (2005)","journal-title":"Math. Methods Appl. Sci."},{"key":"10134_CR8","doi-asserted-by":"crossref","unstructured":"Boi, S., Capasso, V., Morale, D.: Modeling the aggregative behavior of ants of the species polyergus rufescens. Nonlinear Anal. Real World Appl. 1(1), 163\u2013176 (2000). Spatial heterogeneity in ecological models (Alcal\u00e1 de Henares, 1998)","DOI":"10.1016\/S0362-546X(99)00399-5"},{"issue":"4","key":"10134_CR9","doi-asserted-by":"crossref","first-page":"749","DOI":"10.3934\/nhm.2008.3.749","volume":"3","author":"M Burger","year":"2008","unstructured":"Burger, M., Di Francesco, M.: Large time behavior of nonlocal aggregation models with nonlinear diffusion. Netw. Heterog. Media 3(4), 749\u2013785 (2008)","journal-title":"Netw. Heterog. Media"},{"issue":"5","key":"10134_CR10","doi-asserted-by":"crossref","first-page":"1701","DOI":"10.4171\/jems\/401","volume":"15","author":"L Caffarelli","year":"2013","unstructured":"Caffarelli, L., Soria, F., V\u00e1zquez, J.L.: Regularity of solutions of the fractional porous medium flow. J. Eur. Math. Soc. 15(5), 1701\u20131746 (2013)","journal-title":"J. Eur. Math. Soc."},{"issue":"3","key":"10134_CR11","doi-asserted-by":"crossref","first-page":"1529","DOI":"10.1007\/s00205-021-01676-x","volume":"241","author":"JA Carrillo","year":"2021","unstructured":"Carrillo, J.A., Choi, Y.-P.: Mean-field limits: from particle descriptions to macroscopic equations. Arch. Ration. Mech. Anal. 241(3), 1529\u20131573 (2021)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"2","key":"10134_CR12","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1142\/S0218202521500081","volume":"31","author":"JA Carrillo","year":"2021","unstructured":"Carrillo, J.A., Choi, Y.-P., Jung, J.: Quantifying the hydrodynamic limit of Vlasov-type equations with alignment and nonlocal forces. Math. Models Methods Appl. Sci. 31(2), 327\u2013408 (2021)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"3","key":"10134_CR13","doi-asserted-by":"crossref","first-page":"355","DOI":"10.3934\/krm.2021052","volume":"15","author":"JA Carrillo","year":"2022","unstructured":"Carrillo, J.A., Choi, Y.-P., Peng, Y.: Large friction-high force fields limit for the nonlinear Vlasov\u2013Poisson\u2013Fokker\u2013Planck system. Kinetic Relat. Models 15(3), 355\u2013384 (2022)","journal-title":"Kinetic Relat. Models"},{"key":"10134_CR14","doi-asserted-by":"crossref","first-page":"02","DOI":"10.1215\/00127094-2010-211","volume":"156","author":"JA Carrillo","year":"2011","unstructured":"Carrillo, J.A., Di Francesco, M., Figalli, A., Laurent, T., Slep\u010dev, D.: Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Duke Math. J. 156, 02 (2011)","journal-title":"Duke Math. J."},{"key":"10134_CR15","first-page":"07","volume":"21","author":"JA Ca\u00f1izo","year":"2009","unstructured":"Ca\u00f1izo, J.A., Carrillo, J.A., Rosado, J.: A well-posedness theory in measures for some kinetic models of collective motion. Math. Models Methods Appl. Sci. 21, 07 (2009)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"10","key":"10134_CR16","doi-asserted-by":"crossref","first-page":"2465","DOI":"10.2140\/apde.2023.16.2465","volume":"16","author":"L Cesbron","year":"2023","unstructured":"Cesbron, L., Iacobelli, M.: Global well-posedness of Vlasov\u2013Poisson-type systems in bounded domains. Anal. PDE 16(10), 2465\u20132494 (2023)","journal-title":"Anal. PDE"},{"key":"10134_CR17","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1016\/j.jde.2021.07.024","volume":"299","author":"Y-P Choi","year":"2021","unstructured":"Choi, Y.-P.: Large friction limit of pressureless Euler equations with nonlocal forces. J. Differ. Equ. 299, 196\u2013228 (2021)","journal-title":"J. Differ. Equ."},{"key":"10134_CR18","doi-asserted-by":"crossref","first-page":"112393","DOI":"10.1016\/j.na.2021.112393","volume":"210","author":"Y-P Choi","year":"2021","unstructured":"Choi, Y.-P., Jeong, I.-J.: Classical solutions for fractional porous medium flow. Nonlinear Anal. 210, 112393 (2021)","journal-title":"Nonlinear Anal."},{"key":"10134_CR19","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/j.jde.2022.05.008","volume":"330","author":"Y-P Choi","year":"2022","unstructured":"Choi, Y.-P., Tse, O.: Quantified overdamped limit for kinetic Vlasov\u2013Fokker\u2013Planck equations with singular interaction forces. J. Differ. Equ. 330, 150\u2013207 (2022)","journal-title":"J. Differ. Equ."},{"issue":"1","key":"10134_CR20","doi-asserted-by":"crossref","first-page":"41","DOI":"10.3934\/krm.2022021","volume":"16","author":"Y-P Choi","year":"2023","unstructured":"Choi, Y.-P., Jeong, I.-J.: Global-in-time existence of weak solutions for Vlasov\u2013Manev\u2013Fokker\u2013Planck system. Kinet. Relat. Models 16(1), 41\u201353 (2023)","journal-title":"Kinet. Relat. Models"},{"issue":"2","key":"10134_CR21","doi-asserted-by":"crossref","first-page":"1525","DOI":"10.1137\/22M1537643","volume":"56","author":"Y-P Choi","year":"2024","unstructured":"Choi, Y.-P., Jung, J.: Modulated energy estimates for singular kernels and their applications to asymptotic analyses for kinetic equations. SIAM J. Math. Anal. 56(2), 1525\u20131559 (2024)","journal-title":"SIAM J. Math. Anal."},{"key":"10134_CR22","doi-asserted-by":"crossref","first-page":"2777","DOI":"10.1088\/0951-7715\/26\/10\/2777","volume":"26","author":"M Di Francesco","year":"2013","unstructured":"Di Francesco, M., Fagioli, S.: Measure solutions for non-local interaction PDEs with two species. Nonlinearity 26, 2777 (2013)","journal-title":"Nonlinearity"},{"issue":"10","key":"10134_CR23","doi-asserted-by":"crossref","first-page":"4517","DOI":"10.1088\/1361-6544\/aaced5","volume":"31","author":"MH Duong","year":"2018","unstructured":"Duong, M.H., Lamacz, A., Peletier, M.A., Schlichting, A., Sharma, U.: Quantification of coarse-graining error in Langevin and overdamped Langevin dynamics. Nonlinearity 31(10), 4517 (2018)","journal-title":"Nonlinearity"},{"issue":"4","key":"10134_CR24","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1007\/s00526-017-1186-9","volume":"56","author":"MH Duong","year":"2017","unstructured":"Duong, M.H., Lamacz, A., Peletier, M.A., Sharma, U.: Variational approach to coarse-graining of generalized gradient flows. Calc. Var. Part. Differ. Equ. 56(4), 100 (2017)","journal-title":"Calc. Var. Part. Differ. Equ."},{"issue":"11","key":"10134_CR25","doi-asserted-by":"crossref","first-page":"6774","DOI":"10.1016\/j.jde.2015.08.018","volume":"259","author":"RC Fetecau","year":"2015","unstructured":"Fetecau, R.C., Sun, W.: First-order aggregation models and zero inertia limits. J. Differ. Equ. 259(11), 6774\u20136802 (2015)","journal-title":"J. Differ. Equ."},{"issue":"3","key":"10134_CR26","doi-asserted-by":"crossref","first-page":"843","DOI":"10.2140\/apde.2019.12.843","volume":"12","author":"A Figalli","year":"2019","unstructured":"Figalli, A., Kang, M.-J.: A rigorous derivation from the kinetic Cucker\u2013Smale model to the pressureless Euler system with nonlocal alignment. Anal. PDE 12(3), 843\u2013866 (2019)","journal-title":"Anal. PDE"},{"issue":"3","key":"10134_CR27","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1007\/s10955-004-2273-9","volume":"117","author":"M Freidlin","year":"2004","unstructured":"Freidlin, M.: Some remarks on the Smoluchowski\u2013Kramers approximation. J. Stat. Phys. 117(3), 617\u2013634 (2004)","journal-title":"J. Stat. Phys."},{"key":"10134_CR28","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611971477","volume-title":"The Cauchy Problem in Kinetic Theory","author":"RT Glassey","year":"1996","unstructured":"Glassey, R.T.: The Cauchy Problem in Kinetic Theory. Society for Industrial and Applied Mathematics (1996)"},{"issue":"8","key":"10134_CR29","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1016\/S0021-7824(99)00021-5","volume":"78","author":"F Golse","year":"1999","unstructured":"Golse, F., Saint-Raymond, L.: The Vlasov\u2013Poisson system with strong magnetic field. J. de Math\u00e9matiques Pures et Appliqu\u00e9es 78(8), 791\u2013817 (1999)","journal-title":"J. de Math\u00e9matiques Pures et Appliqu\u00e9es"},{"issue":"05","key":"10134_CR30","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1142\/S021820250500056X","volume":"15","author":"T Goudon","year":"2005","unstructured":"Goudon, T.: Hydrodynamic limit for the Vlasov\u2013Poisson\u2013Fokker\u2013Planck system: analysis of the two-dimensional case. Math. Models Methods Appl. Sci. 15(05), 737\u2013752 (2005)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"4","key":"10134_CR31","doi-asserted-by":"crossref","first-page":"762","DOI":"10.1007\/s10955-012-0418-9","volume":"146","author":"S Hottovy","year":"2012","unstructured":"Hottovy, S., Volpe, G., Wehr, J.: Noise-induced drift in stochastic differential equations with arbitrary friction and diffusion in the Smoluchowski\u2013Kramers limit. J. Stat. Phys. 146(4), 762\u2013773 (2012)","journal-title":"J. Stat. Phys."},{"issue":"1","key":"10134_CR32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/S0036141096303359","volume":"29","author":"R Jordan","year":"1998","unstructured":"Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker\u2013Planck equation. SIAM J. Math. Anal. 29(1), 1\u201317 (1998)","journal-title":"SIAM J. Math. Anal."},{"issue":"1","key":"10134_CR33","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1137\/120866828","volume":"45","author":"TK Karper","year":"2013","unstructured":"Karper, T.K., Mellet, A., Trivisa, K.: Existence of weak solutions to kinetic flocking models. SIAM J. Math. Anal. 45(1), 215\u2013243 (2013)","journal-title":"SIAM J. Math. Anal."},{"issue":"4","key":"10134_CR34","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1016\/S0031-8914(40)90098-2","volume":"7","author":"HA Kramers","year":"1940","unstructured":"Kramers, H.A.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7(4), 284\u2013304 (1940)","journal-title":"Physica"},{"issue":"3","key":"10134_CR35","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1007\/s00205-004-0307-8","volume":"172","author":"H Li","year":"2004","unstructured":"Li, H., Toscani, G.: Long-time asymptotics of kinetic models of granular flows. Arch. Ration. Mech. Anal. 172(3), 407\u2013428 (2004)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"6","key":"10134_CR36","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1007\/s002850050158","volume":"38","author":"A Mogilner","year":"1999","unstructured":"Mogilner, A., Edelstein-Keshet, L.: A non-local model for a swarm. J. Math. Biol. 38(6), 534\u2013570 (1999)","journal-title":"J. Math. Biol."},{"issue":"1","key":"10134_CR37","first-page":"41","volume":"42","author":"K Narita","year":"1994","unstructured":"Narita, K.: Asymptotic behavior of fluctuation and deviation from limit system in the Smoluchowski\u2013Kramers approximation for SDE. Yokohama Math. J. 42(1), 41\u201376 (1994)","journal-title":"Yokohama Math. J."},{"key":"10134_CR38","first-page":"45","volume":"4","author":"H Nguyen","year":"2022","unstructured":"Nguyen, H., Rosenzweig, Q., Serfaty, S.: Mean-field limits of Riesz-type singular flows. Ars Inven. Anal. 4, 45 (2022)","journal-title":"Ars Inven. Anal."},{"key":"10134_CR39","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-4978-6","volume-title":"Diffusion and Ecological Problems: Modern Perspectives, Volume 14 of Interdisciplinary Applied Mathematics","author":"A Okubo","year":"2001","unstructured":"Okubo, A., Levin, S.A.: Diffusion and Ecological Problems: Modern Perspectives, Volume 14 of Interdisciplinary Applied Mathematics. Springer, New York (2001)"},{"issue":"07","key":"10134_CR40","doi-asserted-by":"crossref","first-page":"1027","DOI":"10.1142\/S0218202500000525","volume":"10","author":"F Poupaud","year":"2000","unstructured":"Poupaud, F., Soler, J.: Parabolic limit and stability of the Vlasov\u2013Fokker\u2013Planck system. Math. Models Methods Appl. Sci. 10(07), 1027\u20131045 (2000)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"1","key":"10134_CR41","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1214\/20-AIHP1087","volume":"57","author":"M R\u00f6ckner","year":"2021","unstructured":"R\u00f6ckner, M., Sun, X., Xie, Y.: Strong convergence order for slow-fast McKean\u2013Vlasov stochastic differential equations. Ann. Inst. Henri Poincar\u00e9 Probab. Stat. 57(1), 547\u2013576 (2021)","journal-title":"Ann. Inst. Henri Poincar\u00e9 Probab. Stat."},{"key":"10134_CR42","first-page":"575","volume":"31","author":"AN Tikhonov","year":"1952","unstructured":"Tikhonov, A.N.: Systems of differential equations containing a small parameter multiplying the derivative. Mathematical Sbornik 31, 575\u2013586 (1952)","journal-title":"Mathematical Sbornik"},{"issue":"1","key":"10134_CR43","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1137\/S0036139903437424","volume":"65","author":"CM Topaz","year":"2004","unstructured":"Topaz, C.M., Bertozzi, A.L.: Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM J. Appl. Math. 65(1), 152\u2013174 (2004)","journal-title":"SIAM J. Appl. Math."},{"key":"10134_CR44","volume-title":"Topics in Optimal Transportation, Volume 58 of Graduate Studies in Mathematics","author":"C Villani","year":"2003","unstructured":"Villani, C.: Topics in Optimal Transportation, Volume 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2003)"},{"key":"10134_CR45","doi-asserted-by":"crossref","unstructured":"Villani, C.: Optimal Transport, Volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2009)","DOI":"10.1007\/978-3-540-71050-9"}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-025-10134-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-025-10134-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-025-10134-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,13]],"date-time":"2025-03-13T07:41:39Z","timestamp":1741851699000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-025-10134-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,9]]},"references-count":45,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,4]]}},"alternative-id":["10134"],"URL":"https:\/\/doi.org\/10.1007\/s00332-025-10134-x","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,9]]},"assertion":[{"value":"5 February 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 January 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 February 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"39"}}