{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,21]],"date-time":"2026-03-21T20:51:39Z","timestamp":1774126299005,"version":"3.50.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2021,7,8]],"date-time":"2021-07-08T00:00:00Z","timestamp":1625702400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,7,8]],"date-time":"2021-07-08T00:00:00Z","timestamp":1625702400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100010198","name":"Ministerio de Econom\u00eda, Industria y Competitividad, Gobierno de Espa\u00f1a","doi-asserted-by":"publisher","award":["RTI2018\u2013098850-B\u2013I00"],"award-info":[{"award-number":["RTI2018\u2013098850-B\u2013I00"]}],"id":[{"id":"10.13039\/501100010198","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100014440","name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","doi-asserted-by":"publisher","award":["FPU15\/04094"],"award-info":[{"award-number":["FPU15\/04094"]}],"id":[{"id":"10.13039\/100014440","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011011","name":"Junta de Andaluc\u00eda","doi-asserted-by":"publisher","award":["FQM 954"],"award-info":[{"award-number":["FQM 954"]}],"id":[{"id":"10.13039\/501100011011","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011011","name":"Junta de Andaluc\u00eda","doi-asserted-by":"publisher","award":["P18-RT-2422"],"award-info":[{"award-number":["P18-RT-2422"]}],"id":[{"id":"10.13039\/501100011011","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2021,10]]},"DOI":"10.1007\/s00332-021-09729-x","type":"journal-article","created":{"date-parts":[[2021,7,8]],"date-time":"2021-07-08T09:02:41Z","timestamp":1625734961000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["Vortex Patches Choreography for Active Scalar Equations"],"prefix":"10.1007","volume":"31","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9016-3636","authenticated-orcid":false,"given":"Claudia","family":"Garc\u00eda","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,7,8]]},"reference":[{"key":"9729_CR1","unstructured":"Ao, W., D\u00e1vila, J., Del Pino, M., Musso, M., Wei, J.: Travelling and rotating solutions to the generalized inviscid surface quasi\u2013geostrophic equation. arXiv:2008.12911"},{"issue":"1","key":"9729_CR2","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1007\/BF02097055","volume":"152","author":"AL Bertozzi","year":"1993","unstructured":"Bertozzi, A.L., Constantin, P.: Global regularity for vortex patches. Comm. Math. Phys. 152(1), 19\u201328 (1993)","journal-title":"Comm. Math. Phys."},{"key":"9729_CR3","doi-asserted-by":"publisher","first-page":"2051","DOI":"10.1016\/j.physd.2008.02.030","volume":"237","author":"S Boatto","year":"2008","unstructured":"Boatto, S., Sim\u00f3, C.: Thomson\u2019s Heptagon: A case of bifurcation at infinity. Phys. D 237, 2051\u20132055 (2008)","journal-title":"Phys. D"},{"key":"9729_CR4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF02281165","volume":"6","author":"J Burbea","year":"1982","unstructured":"Burbea, J.: Motions of vortex patches. Lett. Math. Phys. 6, 1\u201316 (1982)","journal-title":"Lett. Math. Phys."},{"key":"9729_CR5","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1134\/S156035471805009X","volume":"23","author":"RC Calleja","year":"2018","unstructured":"Calleja, R.C., Doedel, E.J., Garc\u00eda-azpeitia, C.: Choreographies in the n-vortex problem. Regul. Chaot. Dyn 23, 595\u2013612 (2018)","journal-title":"Regul. Chaot. Dyn"},{"issue":"5","key":"9729_CR6","doi-asserted-by":"publisher","first-page":"935","DOI":"10.1215\/00127094-3449673","volume":"165","author":"A Castro","year":"2016","unstructured":"Castro, A., C\u00f3rdoba, D., G\u00f3mez-Serrano, J.: Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations. Duke Math. J. 165(5), 935\u2013984 (2016)","journal-title":"Duke Math. J."},{"issue":"1","key":"9729_CR7","first-page":"34","volume":"2","author":"A Castro","year":"2016","unstructured":"Castro, A., C\u00f3rdoba, D., G\u00f3mez-Serrano, J.: Uniformly rotating analytic global patch solutions for active scalars. J. Ann. PDE 2(1), 34 (2016). (Art. 1)","journal-title":"J. Ann. PDE"},{"issue":"2","key":"9729_CR8","doi-asserted-by":"publisher","first-page":"719","DOI":"10.1007\/s00205-018-1288-3","volume":"231","author":"A Castro","year":"2019","unstructured":"Castro, A., C\u00f3rdoba, D., G\u00f3mez-Serrano, J.: Uniformly rotating smooth solutions for the incompressible 2D Euler equations. Arch. Ration. Mech. Anal. 231(2), 719\u2013785 (2019)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"9729_CR9","unstructured":"Castro, A., C\u00f3rdoba, D., G\u00f3mez-Serrano, J.: Global smooth solutions for the inviscid SQG equation, arXiv:1603.03325, (2016)"},{"key":"9729_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.24033\/asens.1665","volume":"26","author":"J-Y Chemin","year":"1993","unstructured":"Chemin, J.-Y.: Persistance de structures g\u00e9ometriques dans les fluides incompressibles bidimensionnels. Ann. Sci. Ec. Norm. Sup. 26, 1\u201326 (1993)","journal-title":"Ann. Sci. Ec. Norm. Sup."},{"issue":"6","key":"9729_CR11","doi-asserted-by":"publisher","first-page":"1495","DOI":"10.1088\/0951-7715\/7\/6\/001","volume":"7","author":"P Constantin","year":"1994","unstructured":"Constantin, P., Majda, A.J., Tabak, E.: Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7(6), 1495\u20131533 (1994)","journal-title":"Nonlinearity"},{"key":"9729_CR12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/btran\/20","volume":"5","author":"A C\u00f3rdoba","year":"2018","unstructured":"C\u00f3rdoba, A., C\u00f3rdoba, D., Gancedo, F.: Uniqueness for SQG patch solutions. Trans. Amer. Math. Soc. Ser. B 5, 1\u201331 (2018)","journal-title":"Trans. Amer. Math. Soc. Ser. B"},{"issue":"17","key":"9729_CR13","doi-asserted-by":"publisher","first-page":"5949","DOI":"10.1073\/pnas.0501977102","volume":"102","author":"D C\u00f3rdoba","year":"2005","unstructured":"C\u00f3rdoba, D., Fontelos, M.A., Mancho, A.M., Rodrigo, J.L.: Evidence of singularities for a family of contour dynamics equations. Proc. Natl. Acad. Sci. USA 102(17), 5949\u20135952 (2005)","journal-title":"Proc. Natl. Acad. Sci. USA"},{"issue":"3","key":"9729_CR14","doi-asserted-by":"publisher","first-page":"1467","DOI":"10.1007\/s00205-019-01448-8","volume":"235","author":"J Davila","year":"2020","unstructured":"Davila, J., Del Pino, M., Musso, M., Wei, J.: Gluing Methods for Vortex Dynamics in Euler Flows. Arch. Ration. Mech. Anal. 235(3), 1467\u20131530 (2020)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"9729_CR15","doi-asserted-by":"crossref","unstructured":"Dritschel, D.G., Hmidi, T., Renault, C.: Imperfect bifurcation for the quasi-geostrophic shallow-water equations, arXiv:1801.02092, (2018)","DOI":"10.1007\/s00205-018-1312-7"},{"key":"9729_CR16","doi-asserted-by":"publisher","first-page":"1209","DOI":"10.1007\/s00205-015-0953-z","volume":"220","author":"F De la Hoz","year":"2016","unstructured":"De la Hoz, F., Hassainia, Z., Hmidi, T.: Doubly Connected V-States for the Generalized Surface Quasi-Geostrophic Equations. Arch. Ration. Mech. Anal 220, 1209\u20131281 (2016)","journal-title":"Arch. Ration. Mech. Anal"},{"key":"9729_CR17","doi-asserted-by":"publisher","first-page":"1892","DOI":"10.1137\/140992801","volume":"48","author":"F De la Hoz","year":"2016","unstructured":"De la Hoz, F., Hmidi, T., Mateu, J., Verdera, J.: Doubly connected V-states for the planar Euler equations. SIAM J. Math. Anal. 48, 1892\u20131928 (2016)","journal-title":"SIAM J. Math. Anal."},{"key":"9729_CR18","doi-asserted-by":"publisher","first-page":"859","DOI":"10.1103\/PhysRevLett.40.859","volume":"40","author":"GS Deem","year":"1978","unstructured":"Deem, G.S., Zabusky, N.J.: Vortex waves: stationary \u201cV-states\u2019\u2019, Interactions, Recurrence, and Breaking. Phys. Rev. Lett. 40, 859\u2013862 (1978)","journal-title":"Phys. Rev. Lett."},{"issue":"6","key":"9729_CR19","doi-asserted-by":"publisher","first-page":"2569","DOI":"10.1016\/j.aim.2007.10.010","volume":"217","author":"F Gancedo","year":"2008","unstructured":"Gancedo, F.: Existence for the $$\\alpha $$-patch model and the QG sharp front in Sobolev spaces. Adv. Math. 217(6), 2569\u20132598 (2008)","journal-title":"Adv. Math."},{"issue":"4","key":"9729_CR20","doi-asserted-by":"publisher","first-page":"1625","DOI":"10.1088\/1361-6544\/ab6309","volume":"33","author":"C Garc\u00eda","year":"2020","unstructured":"Garc\u00eda, C.: K\u00e1rm\u00e1n Vortex Street in incompressible fluid models. Nonlinearity 33(4), 1625\u20131676 (2020)","journal-title":"Nonlinearity"},{"key":"9729_CR21","unstructured":"Garc\u00eda, C., Hmidi, T., Mateu, J.: Time periodic solutions for 3D quasi\u2013geostrophic model. arXiv:2004.01644, (2020)"},{"key":"9729_CR22","doi-asserted-by":"publisher","first-page":"929","DOI":"10.1007\/s00205-020-01561-z","volume":"238","author":"C Garc\u00eda","year":"2020","unstructured":"Garc\u00eda, C., Hmidi, T., Soler, J.: Non uniform rotating vortices and periodic orbits for the two-dimensional Euler equations. Arch. Ration. Mech. Anal 238, 929\u20131086 (2020)","journal-title":"Arch. Ration. Mech. Anal"},{"key":"9729_CR23","unstructured":"G\u00f3mez-Serrano, J., Park, J., Shi, J., Yao, Y.: Symmetry in stationary and uniformly-rotating solutions of active scalar equations, arXiv:1908.01722, (2019)"},{"issue":"1","key":"9729_CR24","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1007\/s00220-015-2300-5","volume":"337","author":"Z Hassainia","year":"2015","unstructured":"Hassainia, Z., Hmidi, T.: On the V-states for the generalized quasi-geostrophic equations. Comm. Math. Phys. 337(1), 321\u2013377 (2015)","journal-title":"Comm. Math. Phys."},{"key":"9729_CR25","unstructured":"Hassainia, Z., Masmoudi, N., Wheeler, M.H.: Global bifurcation of rotating vortex patches, arXiv:1712.03085, (2017)"},{"key":"9729_CR26","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-84882-319-8","volume-title":"Potential theory","author":"LL Helms","year":"2009","unstructured":"Helms, L.L.: Potential theory. Springer, London (2009)"},{"key":"9729_CR27","doi-asserted-by":"publisher","first-page":"801","DOI":"10.1007\/s00028-015-0281-7","volume":"15","author":"T Hmidi","year":"2015","unstructured":"Hmidi, T.: On the trivial solutions for the vortex patch rotating model. J. Evol. Equ. 15, 801\u2013816 (2015)","journal-title":"J. Evol. Equ."},{"key":"9729_CR28","doi-asserted-by":"publisher","first-page":"5401","DOI":"10.3934\/dcds.2016038","volume":"36","author":"T Hmidi","year":"2016","unstructured":"Hmidi, T., Mateu, J.: Bifurcation of rotating patches from Kirchhoff vortices. Discret. Contin. Dyn. Syst. 36, 5401\u20135422 (2016)","journal-title":"Discret. Contin. Dyn. Syst."},{"issue":"2","key":"9729_CR29","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1007\/s00220-016-2784-7","volume":"350","author":"T Hmidi","year":"2017","unstructured":"Hmidi, T., Mateu, J.: Existence of corotating and counter-rotating vortex pairs for active scalar equations. Comm. Math. Phys. 350(2), 699\u2013747 (2017)","journal-title":"Comm. Math. Phys."},{"key":"9729_CR30","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1007\/s00205-013-0618-8","volume":"209","author":"T Hmidi","year":"2013","unstructured":"Hmidi, T., Mateu, J., Verdera, J.: Boundary regularity of rotating vortex patches. Arch. Ration. Mech. Anal 209, 171\u2013208 (2013)","journal-title":"Arch. Ration. Mech. Anal"},{"key":"9729_CR31","volume-title":"Vorlesungenber mathematische Physik. Mechanik","author":"GR Kirchhoff","year":"1876","unstructured":"Kirchhoff, G.R.: Vorlesungenber mathematische Physik. Mechanik. Teubner, Leipzig (1876)"},{"issue":"7","key":"9729_CR32","doi-asserted-by":"publisher","first-page":"1253","DOI":"10.1002\/cpa.21677","volume":"70","author":"A Kiselev","year":"2017","unstructured":"Kiselev, A., Yao, Y., Zlatos, A.: Local regularity for the modified SQG patch equation. Comm. Pure Appl. Math. 70(7), 1253\u20131315 (2017)","journal-title":"Comm. Pure Appl. Math."},{"key":"9729_CR33","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-9593-2","volume-title":"Linear Integral Equations","author":"R Kress","year":"2014","unstructured":"Kress, R.: Linear Integral Equations. Springer, New York (2014)"},{"key":"9729_CR34","doi-asserted-by":"publisher","first-page":"574","DOI":"10.1063\/1.1482175","volume":"12","author":"LG Kurakin","year":"2020","unstructured":"Kurakin, L.G., Yudovich, V.I.: The stability of stationary rotation of a regular vortex polygon. Chaos Interdiscip J Nonlinear Sci 12, 574\u2013595 (2020)","journal-title":"Chaos Interdiscip J Nonlinear Sci"},{"key":"9729_CR35","volume-title":"Analysis","author":"E Lieb","year":"1997","unstructured":"Lieb, E., Loss, M.: Analysis. American Mathematical Society, Ny (1997)"},{"issue":"4","key":"9729_CR36","doi-asserted-by":"publisher","first-page":"563","DOI":"10.1007\/BF01206023","volume":"91","author":"C Marchioro","year":"1983","unstructured":"Marchioro, C., Pulvirenti, M.: Euler evolution for singular initial data and vortex theory. Comm. Math. Phys. 91(4), 563\u2013572 (1983)","journal-title":"Comm. Math. Phys."},{"key":"9729_CR37","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9290-3","volume-title":"The N-Vortex Problem, Analytical Techniques","author":"PK Newton","year":"2001","unstructured":"Newton, P.K.: The N-Vortex Problem, Analytical Techniques. Springer, New York (2001)"},{"issue":"6","key":"9729_CR38","doi-asserted-by":"publisher","first-page":"821","DOI":"10.1002\/cpa.20059","volume":"58","author":"JL Rodrigo","year":"2005","unstructured":"Rodrigo, J.L.: On the evolution of sharp fronts for the quasi-geostrophic equation. Comm. Pure Appl. Math. 58(6), 821\u2013866 (2005)","journal-title":"Comm. Pure Appl. Math."},{"issue":"6","key":"9729_CR39","first-page":"515","volume":"318","author":"P Serfati","year":"1994","unstructured":"Serfati, P.: Une preuve directe d\u2019existence globale des vortex patches 2D. C. R. Acad. Sci. Paris S\u00e9r. I Math. 318(6), 515\u2013518 (1994)","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. I Math."},{"issue":"3","key":"9729_CR40","doi-asserted-by":"publisher","first-page":"869","DOI":"10.1007\/s00205-010-0293-y","volume":"198","author":"D Smets","year":"2010","unstructured":"Smets, D., Van Schaftingen, J.: Desingularization of vortices for the Euler equation. Arch. Ration. Mech. Anal. 198(3), 869\u2013925 (2010)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"9729_CR41","first-page":"1032","volume":"3","author":"Y Yudovich","year":"1963","unstructured":"Yudovich, Y.: Nonstationary flow of an ideal incompressible liquid. Zh. Vych. Mat. 3, 1032\u20131066 (1963)","journal-title":"Zh. Vych. Mat."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-021-09729-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-021-09729-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-021-09729-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,20]],"date-time":"2021-09-20T08:07:29Z","timestamp":1632125249000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-021-09729-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,8]]},"references-count":41,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2021,10]]}},"alternative-id":["9729"],"URL":"https:\/\/doi.org\/10.1007\/s00332-021-09729-x","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,8]]},"assertion":[{"value":"26 October 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 June 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 July 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"75"}}