{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,8]],"date-time":"2026-07-08T03:14:58Z","timestamp":1783480498534,"version":"3.55.0"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T00:00:00Z","timestamp":1768867200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T00:00:00Z","timestamp":1768867200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"funder":[{"name":"JSPS KAKENHI","award":["23H00086"],"award-info":[{"award-number":["23H00086"]}]},{"DOI":"10.13039\/501100001809","name":"NNSF of China","doi-asserted-by":"crossref","award":["12301142"],"award-info":[{"award-number":["12301142"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"NSFSC","award":["2024NSFSC1341"],"award-info":[{"award-number":["2024NSFSC1341"]}]},{"name":"CPSF","award":["2022M722286"],"award-info":[{"award-number":["2022M722286"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2026,2]]},"DOI":"10.1007\/s00332-025-10236-6","type":"journal-article","created":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T05:17:24Z","timestamp":1768886244000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Steady Vortex Patches on Flat Torus with a Constant Background Vorticity"],"prefix":"10.1007","volume":"36","author":[{"given":"Takashi","family":"Sakajo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Changjun","family":"Zou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,1,20]]},"reference":[{"key":"10236_CR1","unstructured":"Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume No. 55 of National Bureau of Standards Applied Mathematics Series. U. S. Government Printing Office, Washington, DC, For sale by the Superintendent of Documents (1964)"},{"issue":"9","key":"10236_CR2","doi-asserted-by":"publisher","first-page":"6665","DOI":"10.1090\/tran\/8406","volume":"374","author":"W Ao","year":"2021","unstructured":"Ao, W., D\u00e1vila, J., del Pino, M., Musso, M., Wei, J.: Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation. Trans. Am. Math. Soc. 374(9), 6665\u20136689 (2021)","journal-title":"Trans. Am. Math. Soc."},{"issue":"5","key":"10236_CR3","doi-asserted-by":"publisher","first-page":"1002","DOI":"10.1016\/0021-8928(65)90119-X","volume":"29","author":"VI Arnol\u2019d","year":"1965","unstructured":"Arnol\u2019d, V.I.: Variational principle for three-dimensional steady-state flows of an ideal fluid. J. Appl. Math. Mech. 29(5), 1002\u20131008 (1965)","journal-title":"J. Appl. Math. Mech."},{"issue":"15","key":"10236_CR4","doi-asserted-by":"publisher","first-page":"5125","DOI":"10.1088\/0305-4470\/20\/15\/028","volume":"20","author":"R Benzi","year":"1987","unstructured":"Benzi, R., Legras, B.: Wave-vortex dynamics. J. Phys. A: Math. Gen. 20(15), 5125 (1987)","journal-title":"J. Phys. A: Math. Gen."},{"issue":"3","key":"10236_CR5","doi-asserted-by":"publisher","first-page":"1279","DOI":"10.1007\/s00222-023-01195-4","volume":"233","author":"M Berti","year":"2023","unstructured":"Berti, M., Hassainia, Z., Masmoudi, N.: Time quasi-periodic vortex patches of Euler equation in the plane. Invent. Math. 233(3), 1279\u20131391 (2023)","journal-title":"Invent. Math."},{"issue":"1","key":"10236_CR6","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), 1\u201316 (1982)","journal-title":"Lett. Math. Phys."},{"key":"10236_CR7","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1016\/j.aim.2014.09.027","volume":"270","author":"D Cao","year":"2015","unstructured":"Cao, D., Peng, S., Yan, S.: Planar vortex patch problem in incompressible steady flow. Adv. Math. 270, 263\u2013301 (2015)","journal-title":"Adv. Math."},{"key":"10236_CR8","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1016\/j.jde.2021.07.026","volume":"299","author":"D Cao","year":"2021","unstructured":"Cao, D., Qin, G., Zhan, W., Zou, C.: Existence and regularity of co-rotating and traveling-wave vortex solutions for the generalized SQG equation. J. Differ. Equ. 299, 429\u2013462 (2021)","journal-title":"J. Differ. Equ."},{"key":"10236_CR9","doi-asserted-by":"crossref","unstructured":"Cao, D., Qin, G., Zhan, W., Zou, C.: On the global classical solutions for the generalized SQG equation. J. Funct. Anal. 283(2), 109503, 37 (2022)","DOI":"10.1016\/j.jfa.2022.109503"},{"issue":"3","key":"10236_CR10","doi-asserted-by":"publisher","first-page":"2181","DOI":"10.1090\/tran\/8835","volume":"376","author":"D Cao","year":"2023","unstructured":"Cao, D., Qin, G., Zhan, W., Zou, C.: Global solutions for the generalized SQG equation and rearrangements. Trans. Am. Math. Soc. 376(3), 2181\u20132211 (2023)","journal-title":"Trans. Am. Math. Soc."},{"key":"10236_CR11","doi-asserted-by":"crossref","unstructured":"Cao, D., Qin, G., Zhan, W., Zou, C.: K\u00e1rm\u00e1n vortex street for the generalized surface quasi-geostrophic equation. Calc. Var. Partial Differ. Equ. 62(6), 168, 35 (2023)","DOI":"10.1007\/s00526-023-02518-2"},{"issue":"6","key":"10236_CR12","doi-asserted-by":"publisher","first-page":"5363","DOI":"10.1137\/19M1292151","volume":"52","author":"D Cao","year":"2020","unstructured":"Cao, D., Wang, G., Zhan, W.: Desingularization of vortices for two-dimensional steady Euler flows via the vorticity method. SIAM J. Math. Anal. 52(6), 5363\u20135388 (2020)","journal-title":"SIAM J. Math. Anal."},{"issue":"5","key":"10236_CR13","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."},{"key":"10236_CR14","doi-asserted-by":"crossref","unstructured":"Castro, A., C\u00f3rdoba, D., G\u00f3mez-Serrano, J.: Uniformly rotating analytic global patch solutions for active scalars. Ann. PDE 2(1), 1, 34 (2016)","DOI":"10.1007\/s40818-016-0007-3"},{"issue":"2","key":"10236_CR15","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":"10236_CR16","doi-asserted-by":"crossref","unstructured":"Castro, A., C\u00f3rdoba, D., G\u00f3mez-Serrano, J.: Global smooth solutions for the inviscid SQG equation. Mem. Am. Math. Soc. 266(1292), v+89 (2020)","DOI":"10.1090\/memo\/1292"},{"issue":"1","key":"10236_CR17","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1016\/j.aml.2009.07.022","volume":"23","author":"D Crowdy","year":"2010","unstructured":"Crowdy, D.: On rectangular vortex lattices. Appl. Math. Lett. 23(1), 34\u201338 (2010)","journal-title":"Appl. Math. Lett."},{"key":"10236_CR18","doi-asserted-by":"crossref","unstructured":"Crowdy, D.: Solving problems in multiply connected domains, volume\u00a097 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2020)","DOI":"10.1137\/1.9781611976151"},{"key":"10236_CR19","doi-asserted-by":"crossref","unstructured":"D\u00e1vila, J., del Pino, M., Musso, M., Wei, J.: Travelling helices and the vortex filament conjecture in the incompressible Euler equations. Calc. Var. Partial Differ. Equ. 61(4), 119, 30 (2022)","DOI":"10.1007\/s00526-022-02217-4"},{"issue":"3","key":"10236_CR20","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(3), 1209\u20131281 (2016)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"3","key":"10236_CR21","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(3), 1892\u20131928 (2016)","journal-title":"SIAM J. Math. Anal."},{"issue":"3","key":"10236_CR22","doi-asserted-by":"publisher","first-page":"553","DOI":"10.1090\/S0894-0347-1991-1102579-6","volume":"4","author":"J-M Delort","year":"1991","unstructured":"Delort, J.-M.: Existence de nappes de tourbillon en dimension deux. J. Am. Math. Soc. 4(3), 553\u2013586 (1991)","journal-title":"J. Am. Math. Soc."},{"issue":"3","key":"10236_CR23","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1002\/cpa.3160400304","volume":"40","author":"RJ DiPerna","year":"1987","unstructured":"DiPerna, R.J., Majda, A.J.: Concentrations in regularizations for $$2$$-D incompressible flow. Comm. Pure Appl. Math. 40(3), 301\u2013345 (1987)","journal-title":"Comm. Pure Appl. Math."},{"key":"10236_CR24","doi-asserted-by":"crossref","unstructured":"Fraenkel, L.E., Benjamin, T.B.: On steady vortex rings of small cross-section in an ideal fluid. Proc. Roy. Soc. Lond. A Math. Phys. Sci. 316(1524), 29\u201362 (1970)","DOI":"10.1098\/rspa.1970.0065"},{"issue":"1","key":"10236_CR25","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."},{"issue":"9","key":"10236_CR26","doi-asserted-by":"publisher","first-page":"1933","DOI":"10.1002\/cpa.21855","volume":"73","author":"Z Hassainia","year":"2020","unstructured":"Hassainia, Z., Masmoudi, N., Wheeler, M.H.: Global bifurcation of rotating vortex patches. Comm. Pure Appl. Math. 73(9), 1933\u20131980 (2020)","journal-title":"Comm. Pure Appl. Math."},{"key":"10236_CR27","doi-asserted-by":"crossref","unstructured":"Hill, M.J.M.: On a spherical vortex. Philos. Trans. Roy. Soc. A Math. Phys. Eng. Sci 185, 213\u2013245 (1894)","DOI":"10.1098\/rsta.1894.0006"},{"issue":"2","key":"10236_CR28","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."},{"issue":"1","key":"10236_CR29","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(1), 171\u2013208 (2013)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"1","key":"10236_CR30","doi-asserted-by":"publisher","first-page":"727","DOI":"10.1007\/BF01474611","volume":"37","author":"E H\u00f6lder","year":"1933","unstructured":"H\u00f6lder, E.: \u00fcber die unbeschr\u00e4nkte Fortsetzbarkeit einer stetigen ebenen Bewegung in einer unbegrenzten inkompressiblen Fl\u00fcssigkeit. Math. Z. 37(1), 727\u2013738 (1933)","journal-title":"Math. Z."},{"key":"10236_CR31","doi-asserted-by":"crossref","unstructured":"Jim\u00e9nez, J., Guegan, A.: Spontaneous generation of vortex crystals from forced two-dimensional homogeneous turbulence. Phys. Fluids 19(8), 085103, 08 (2007)","DOI":"10.1063\/1.2757713"},{"key":"10236_CR32","unstructured":"Kirchhoff, G.R.: Vorlesungen \u00fcber mathematische physik; bd. 3. Leipzig, (1984)"},{"key":"10236_CR33","doi-asserted-by":"crossref","unstructured":"Krishnamurthy, V.S., Sakajo, T.: The $$N$$-vortex problem in a doubly periodic rectangular domain with constant background vorticity. Physica D 448, 133728, 14 (2023)","DOI":"10.1016\/j.physd.2023.133728"},{"key":"10236_CR34","unstructured":"Lamb, H.: Hydrodynamics, 6th edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1993). With a foreword by R. A. Caflisch [Russel E. Caflisch]"},{"key":"10236_CR35","doi-asserted-by":"crossref","unstructured":"Lin, C.C.: On the motion of vortices in two dimensions. I. Existence of the Kirchhoff-Routh function. Proc. Nat. Acad. Sci. U.S.A 27, 570\u2013575 (1941)","DOI":"10.1073\/pnas.27.12.570"},{"key":"10236_CR36","doi-asserted-by":"crossref","unstructured":"Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics, vol. 27. Cambridge University Press, Cambridge (2002)","DOI":"10.1115\/1.1483363"},{"key":"10236_CR37","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1017\/S0022112084001750","volume":"146","author":"JC Mcwilliams","year":"1984","unstructured":"Mcwilliams, J.C.: The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 21\u201343 (1984)","journal-title":"J. Fluid Mech."},{"issue":"9","key":"10236_CR38","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1016\/0375-9601(87)90053-3","volume":"124","author":"KA O\u2019Neil","year":"1987","unstructured":"O\u2019Neil, K.A.: Symmetric configurations of vortices. Phys. Lett. A 124(9), 503\u2013507 (1987)","journal-title":"Phys. Lett. A"},{"issue":"6","key":"10236_CR39","doi-asserted-by":"publisher","first-page":"1373","DOI":"10.1063\/1.528605","volume":"30","author":"KA O\u2019Neil","year":"1989","unstructured":"O\u2019Neil, K.A.: On the Hamiltonian dynamics of vortex lattices. J. Math. Phys. 30(6), 1373\u20131379 (1989)","journal-title":"J. Math. Phys."},{"issue":"3","key":"10236_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":"10236_CR41","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1017\/S002211209900542X","volume":"392","author":"MA Stremler","year":"1999","unstructured":"Stremler, M.A., Aref, H.: Motion of three point vortices in a periodic parallelogram. J. Fluid Mech. 392, 101\u2013128 (1999)","journal-title":"J. Fluid Mech."},{"key":"10236_CR42","volume-title":"On vortex lattices","author":"VK Tkachenko","year":"1966","unstructured":"Tkachenko, V.K.: On vortex lattices. J. Exp. Theor, Phys (1966)"},{"issue":"9","key":"10236_CR43","doi-asserted-by":"publisher","first-page":"999","DOI":"10.1080\/03605308308820293","volume":"8","author":"B Turkington","year":"1983","unstructured":"Turkington, B.: On steady vortex flow in two dimensions. I. Comm. Partial Differ. Equ. 8(9), 999\u20131030 (1983)","journal-title":"I. Comm. Partial Differ. Equ."},{"issue":"9","key":"10236_CR44","doi-asserted-by":"publisher","first-page":"1031","DOI":"10.1080\/03605308308820294","volume":"8","author":"B Turkington","year":"1983","unstructured":"Turkington, B.: On steady vortex flow in two dimensions. II. Comm. Partial Differ. Equ. 8(9), 1031\u20131071 (1983)","journal-title":"II. Comm. Partial Differ. Equ."},{"key":"10236_CR45","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1016\/j.cam.2017.10.023","volume":"332","author":"T Uda","year":"2018","unstructured":"Uda, T.: Shape calculus for vortex patch equilibria and its application to lattice configurations. J. Comput. Appl. Math. 332, 86\u2013100 (2018)","journal-title":"J. Comput. Appl. Math."},{"issue":"5","key":"10236_CR46","doi-asserted-by":"publisher","first-page":"835","DOI":"10.1063\/1.858014","volume":"3","author":"JB Weiss","year":"1991","unstructured":"Weiss, J.B., McWilliams, J.C.: Nonergodicity of point vortices. Phys. Fluids A 3(5), 835\u2013844 (1991)","journal-title":"Phys. Fluids A"},{"issue":"1","key":"10236_CR47","doi-asserted-by":"publisher","first-page":"698","DOI":"10.1007\/BF01474610","volume":"37","author":"W Wolibner","year":"1933","unstructured":"Wolibner, W.: Un theor\u00e8me sur l\u2019existence du mouvement plan d\u2019un fluide parfait, homog\u00e8ne, incompressible, pendant un temps infiniment long. Math. Z. 37(1), 698\u2013726 (1933)","journal-title":"Math. Z."},{"issue":"6","key":"10236_CR48","doi-asserted-by":"publisher","first-page":"1407","DOI":"10.1016\/0041-5553(63)90247-7","volume":"3","author":"VI Yudovich","year":"1963","unstructured":"Yudovich, V.I.: Non-stationary flow of an ideal incompressible liquid. USSR Comput. Math. Math. Phys. 3(6), 1407\u20131456 (1963)","journal-title":"USSR Comput. Math. Math. Phys."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-025-10236-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-025-10236-6","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-025-10236-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,8]],"date-time":"2026-07-08T02:26:04Z","timestamp":1783477564000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-025-10236-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,20]]},"references-count":48,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,2]]}},"alternative-id":["10236"],"URL":"https:\/\/doi.org\/10.1007\/s00332-025-10236-6","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,20]]},"assertion":[{"value":"13 May 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 December 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 January 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"On behalf of all authors, the corresponding author states that there is no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"20"}}