{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,10]],"date-time":"2026-05-10T00:11:52Z","timestamp":1778371912735,"version":"3.51.4"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2013,6,22]],"date-time":"2013-06-22T00:00:00Z","timestamp":1371859200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2013,12]]},"DOI":"10.1007\/s00332-013-9175-4","type":"journal-article","created":{"date-parts":[[2013,6,21]],"date-time":"2013-06-21T20:08:39Z","timestamp":1371845319000},"page":"993-1000","source":"Crossref","is-referenced-by-count":10,"title":["The 3D Incompressible Euler Equations with a Passive Scalar: A Road to Blow-Up?"],"prefix":"10.1007","volume":"23","author":[{"given":"John D.","family":"Gibbon","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Edriss S.","family":"Titi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2013,6,22]]},"reference":[{"key":"9175_CR1","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-1693-1","volume-title":"Mathematical Methods of Classical Mechanics","author":"V.I. Arnold","year":"1978","unstructured":"Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1978)"},{"key":"9175_CR2","doi-asserted-by":"crossref","DOI":"10.1007\/b97593","volume-title":"Topological Methods in Hydrodynamics","author":"V.I. Arnold","year":"1998","unstructured":"Arnold, V.I., Khesin, B.: Topological Methods in Hydrodynamics. Springer, Berlin (1998)"},{"key":"9175_CR3","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1070\/RM2007v062n03ABEH004410","volume":"62","author":"C. Bardos","year":"2007","unstructured":"Bardos, C., Titi, E.S.: Euler equations of incompressible ideal fluids. Russ. Math. Surv. 62, 409\u2013451 (2007)","journal-title":"Russ. Math. Surv."},{"key":"9175_CR4","first-page":"187","volume":"3","author":"C. Bardos","year":"2010","unstructured":"Bardos, C., Titi, E.S.: Loss of smoothness and energy conserving rough weak solutions for the 3D Euler equations. Discrete Contin. Dyn. Syst. 3, 187\u2013195 (2010)","journal-title":"Discrete Contin. Dyn. Syst."},{"key":"9175_CR5","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1080\/14685248.2013.771838","volume":"14","author":"C. Bardos","year":"2013","unstructured":"Bardos, C., Titi, E.S.: Mathematics and turbulence: where do we stand? J.\u00a0Turbul. 14, 42\u201376 (2013)","journal-title":"J.\u00a0Turbul."},{"issue":"15","key":"9175_CR6","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1016\/j.crma.2012.09.005","volume":"350","author":"C. Bardos","year":"2012","unstructured":"Bardos, C., Titi, E.S., Wiedemann, E.: The vanishing viscosity as a selection principle for the Euler equations: the case of 3D shear flow. C. R. Math. Acad. Sci. Paris, S\u00e9r. I, Math. 350(15), 757\u2013760 (2012)","journal-title":"C. R. Math. Acad. Sci. Paris, S\u00e9r. I, Math."},{"key":"9175_CR7","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/BF01212349","volume":"94","author":"J.T. Beale","year":"1984","unstructured":"Beale, J.T., Kato, T., Majda, A.J.: Remarks on the breakdown of smooth solutions for the 3D Euler equations. Commun. Math. Phys. 94, 61\u201366 (1984)","journal-title":"Commun. Math. Phys."},{"key":"9175_CR8","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1002\/(SICI)1097-0312(199904)52:4<411::AID-CPA1>3.0.CO;2-3","volume":"52","author":"Y. Brenier","year":"1999","unstructured":"Brenier, Y.: Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Commun. Pure Appl. Math. 52, 411\u2013452 (1999)","journal-title":"Commun. Pure Appl. Math."},{"key":"9175_CR9","doi-asserted-by":"crossref","first-page":"1912","DOI":"10.1016\/j.physd.2008.02.007","volume":"237","author":"M.D. Bustamante","year":"2008","unstructured":"Bustamante, M.D., Kerr, R.M.: 3D Euler in a 2D symmetry plane. Physica D 237, 1912\u20131920 (2008)","journal-title":"Physica D"},{"key":"9175_CR10","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1137\/1036004","volume":"36","author":"P. Constantin","year":"1994","unstructured":"Constantin, P.: Geometric statistics in turbulence. SIAM Rev. 36, 73\u201398 (1994)","journal-title":"SIAM Rev."},{"key":"9175_CR11","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1090\/S0273-0979-07-01184-6","volume":"44","author":"P. Constantin","year":"2007","unstructured":"Constantin, P.: On the Euler equations of incompressible fluids. Bull. Am. Math. Soc. 44, 603\u2013621 (2007)","journal-title":"Bull. Am. Math. Soc."},{"key":"9175_CR12","doi-asserted-by":"crossref","first-page":"3307","DOI":"10.1103\/PhysRevE.47.3307","volume":"47","author":"P. Constantin","year":"1993","unstructured":"Constantin, P., Procaccia, I.: Scaling in fluid turbulence: a geometric theory. Phys. Rev. E 47, 3307\u20133315 (1993)","journal-title":"Phys. Rev. E"},{"key":"9175_CR13","doi-asserted-by":"crossref","first-page":"1739","DOI":"10.1103\/PhysRevLett.67.1739","volume":"67","author":"P. Constantin","year":"1991","unstructured":"Constantin, P., Procaccia, I., Sreenivasan, K.R.: Fractal geometry of isoscalar surfaces in turbulence: theory and experiments. Phys. Rev. Lett. 67, 1739\u20131742 (1991)","journal-title":"Phys. Rev. Lett."},{"key":"9175_CR14","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1080\/03605309608821197","volume":"21","author":"P. Constantin","year":"1996","unstructured":"Constantin, P., Fefferman, C., Madja, A.J.: Geometric constraints on potential singular solutions for the 3\u2212D Euler equation. Commun. Partial Differ. Equ. 21, 559\u2013571 (1996)","journal-title":"Commun. Partial Differ. Equ."},{"issue":"3","key":"9175_CR15","doi-asserted-by":"crossref","first-page":"1417","DOI":"10.4007\/annals.2009.170.1417","volume":"170","author":"C. Lellis De","year":"2009","unstructured":"De Lellis, C., Sz\u00e9kelyhidi, L.: The Euler equations as a differential inclusion. Ann. Math. 170(3), 1417\u20131436 (2009)","journal-title":"Ann. Math."},{"key":"9175_CR16","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1007\/s00205-008-0201-x","volume":"195","author":"C. Lellis De","year":"2010","unstructured":"De Lellis, C., Sz\u00e9kelyhidi, L.: On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195, 225\u2013260 (2010)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"9175_CR17","doi-asserted-by":"crossref","first-page":"102","DOI":"10.2307\/1970699","volume":"92","author":"D.G. Ebin","year":"1970","unstructured":"Ebin, D.G., Marsden, J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92, 102\u2013163 (1970)","journal-title":"Ann. Math."},{"key":"9175_CR18","first-page":"271","volume":"59","author":"H. Ertel","year":"1942","unstructured":"Ertel, H.: Ein neuer hydrodynamischer Wirbelsatz. Meteorol. Z. 59, 271\u2013281 (1942)","journal-title":"Meteorol. Z."},{"key":"9175_CR19","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1007\/BF02097394","volume":"155","author":"A.B. Ferrari","year":"1993","unstructured":"Ferrari, A.B.: On the blow-up of solutions of the 3D Euler equations in a bounded domain. Commun. Math. Phys. 155, 277\u2013294 (1993)","journal-title":"Commun. Math. Phys."},{"key":"9175_CR20","doi-asserted-by":"crossref","first-page":"1894","DOI":"10.1016\/j.physd.2007.10.014","volume":"237","author":"J.D. Gibbon","year":"2008","unstructured":"Gibbon, J.D.: The 3D Euler equations: where do we stand? Physica D 237, 1894\u20131904 (2008)","journal-title":"Physica D"},{"key":"9175_CR21","doi-asserted-by":"crossref","first-page":"17200","DOI":"10.1088\/1751-8113\/43\/17\/172001","volume":"43","author":"J.D. Gibbon","year":"2010","unstructured":"Gibbon, J.D., Holm, D.D.: The dynamics of the gradient of potential vorticity. J. Phys. A, Math. Theor. 43, 17200 (2010)","journal-title":"J. Phys. A, Math. Theor."},{"key":"9175_CR22","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1017\/CBO9781139235792.010","volume-title":"Mathematical Aspects of Fluid Mechanics","author":"J.D. Gibbon","year":"2012","unstructured":"Gibbon, J.D., Holm, D.D.: Stretching and folding diagnostics in solutions of the three-dimensional Euler and Navier\u2013Stokes equations. In: Robinson, J.C., Rodrigo, J.L., Sadowski, W. (eds.) Mathematical Aspects of Fluid Mechanics, pp. 201\u2013220. CUP, Cambridge (2012)"},{"key":"9175_CR23","doi-asserted-by":"crossref","first-page":"1932","DOI":"10.1016\/j.physd.2007.11.006","volume":"237","author":"T. Gr\u00e4fke","year":"2008","unstructured":"Gr\u00e4fke, T., Homann, H., Dreher, J., Grauer, R.: Numerical simulations of possible finite time singularities in the incompressible Euler equations. Comparison of numerical methods. Physica D 237, 1932\u20131936 (2008)","journal-title":"Physica D"},{"key":"9175_CR24","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1063\/1.870331","volume":"12","author":"C.R. Graham","year":"2000","unstructured":"Graham, C.R., Henyey, F.: Clebsch representation near points where the vorticity vanishes. Phys. Fluids 12, 744\u2013746 (2000)","journal-title":"Phys. Fluids"},{"key":"9175_CR25","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1002\/qj.49711147002","volume":"111","author":"B.J. Hoskins","year":"1985","unstructured":"Hoskins, B.J., McIntyre, M.E., Robertson, A.W.: On the use and significance of isentropic potential vorticity maps. Q. J. R. Meteorol. Soc. 111, 877\u2013946 (1985)","journal-title":"Q. J. R. Meteorol. Soc."},{"key":"9175_CR26","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1007\/s00332-006-0800-3","volume":"16","author":"T.Y. Hou","year":"2006","unstructured":"Hou, T.Y., Li, R.: Dynamic depletion of vortex stretching and non blow-up of the 3-D incompressible Euler equations. J. Nonlinear Sci. 16, 639\u2013664 (2006)","journal-title":"J. Nonlinear Sci."},{"key":"9175_CR27","doi-asserted-by":"crossref","first-page":"1937","DOI":"10.1016\/j.physd.2008.01.018","volume":"237","author":"T.Y. Hou","year":"2008","unstructured":"Hou, T.Y.: Blow-up or no blow-up? The interplay between theory and numerics. Physica D 237, 1937\u20131944 (2008)","journal-title":"Physica D"},{"key":"9175_CR28","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1016\/0022-1236(72)90003-1","volume":"9","author":"T. Kato","year":"1972","unstructured":"Kato, T.: Non-stationary flows of viscous and ideal flows in R 3. J. Funct. Anal. 9, 296\u2013305 (1972)","journal-title":"J. Funct. Anal."},{"key":"9175_CR29","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/0022-1236(84)90024-7","volume":"56","author":"T. Kato","year":"1984","unstructured":"Kato, T., Lai, C.Y.: Nonlinear evolution equations and the Euler flow. J. Funct. Anal. 56, 15\u201328 (1984)","journal-title":"J. Funct. Anal."},{"key":"9175_CR30","doi-asserted-by":"crossref","first-page":"1725","DOI":"10.1063\/1.858849","volume":"5","author":"R.M. Kerr","year":"1993","unstructured":"Kerr, R.M.: Evidence for a singularity of the three-dimensional incompressible Euler equations. Phys. Fluids A 5, 1725\u20131746 (1993)","journal-title":"Phys. Fluids A"},{"key":"9175_CR31","doi-asserted-by":"crossref","first-page":"822","DOI":"10.1175\/1520-0469(2000)057<0822:MESPVA>2.0.CO;2","volume":"57","author":"M.V. Kurgansky","year":"2000","unstructured":"Kurgansky, M.V., Pisnichenko, I.: Modified Ertel potential vorticity as a climate variable. J. Atmos. Sci. 57, 822\u2013835 (2000)","journal-title":"J. Atmos. Sci."},{"key":"9175_CR32","first-page":"587","volume":"23","author":"M.V. Kurgansky","year":"1987","unstructured":"Kurgansky, M.V., Tatarskaya, M.S.: The potential vorticity concept in meteorology: a review. Izv., Atmos. Ocean. Phys. 23, 587\u2013606 (1987)","journal-title":"Izv., Atmos. Ocean. Phys."},{"key":"9175_CR33","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/BF01506223","volume":"23","author":"L. Lichtenstein","year":"1925","unstructured":"Lichtenstein, L.: Uber einige Existenzproblem der Hydrodynamik homogener unzusammendr\u00fcckbarer, reibunglosser Fl\u00fcssikeiten und die Helmholtzschen Wirbelsalitze. Math. Z. 23, 89\u2013154 (1925)","journal-title":"Math. Z."},{"key":"9175_CR34","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF01475456","volume":"26","author":"L. Lichtenstein","year":"1927","unstructured":"Lichtenstein, L.: Uber einige Existenzproblem der Hydrodynamik homogener unzusammendr\u00fcckbarer, reibunglosser Fl\u00fcssikeiten und die Helmholtzschen Wirbelsalitze. Math. Z. 26, 193\u2013323 (1927)","journal-title":"Math. Z."},{"key":"9175_CR35","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1007\/BF01194657","volume":"32","author":"L. Lichtenstein","year":"1930","unstructured":"Lichtenstein, L.: Uber einige Existenzproblem der Hydrodynamik homogener unzusammendr\u00fcckbarer, reibunglosser Fl\u00fcssikeiten und die Helmholtzschen Wirbelsalitze. Math. Z. 32, 608 (1930)","journal-title":"Math. Z."},{"key":"9175_CR36","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1017\/S0022112069000991","volume":"35","author":"H.K. Moffatt","year":"1969","unstructured":"Moffatt, H.K.: The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117\u2013129 (1969)","journal-title":"J. Fluid Mech."},{"key":"9175_CR37","first-page":"117","volume":"56","author":"S.S. Moiseev","year":"1982","unstructured":"Moiseev, S.S., Sagdeev, R.Z., Tur, A.V., Yanovski, V.V.: On the freezing-in integrals and Lagrange invariants in hydrodynamic models. Sov. Phys. JETP 56, 117\u2013123 (1982)","journal-title":"Sov. Phys. JETP"},{"key":"9175_CR38","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/BF01205787","volume":"98","author":"G. Ponce","year":"1985","unstructured":"Ponce, G.: Remarks on a paper by J.T. Beale, T. Kato, and A. Majda. Commun. Math. Phys. 98, 349\u2013353 (1985)","journal-title":"Commun. Math. Phys."},{"key":"9175_CR39","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1007\/BF02921318","volume":"3","author":"V. Scheffer","year":"1993","unstructured":"Scheffer, V.: An inviscid flow with compact support in space-time. J. Geom. Anal. 3, 343\u2013401 (1993)","journal-title":"J. Geom. Anal."},{"key":"9175_CR40","doi-asserted-by":"crossref","first-page":"1260","DOI":"10.1002\/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6","volume":"50","author":"A. Shnirelman","year":"1997","unstructured":"Shnirelman, A.: On the non-uniqueness of weak solution of the Euler equation. Commun. Pure Appl. Math. 50, 1260\u20131286 (1997)","journal-title":"Commun. Pure Appl. Math."},{"key":"9175_CR41","doi-asserted-by":"crossref","first-page":"77","DOI":"10.3792\/pjaa.69.77","volume":"69","author":"T. Shirota","year":"1993","unstructured":"Shirota, T., Yanagisawa, T.: A continuation principle for the 3D Euler equations for incompressible fluids in a bounded domain. Proc. Jpn. Acad. 69, 77\u201382 (1993)","journal-title":"Proc. Jpn. Acad."},{"issue":"1","key":"9175_CR42","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/0022-1236(75)90052-X","volume":"20","author":"R. Temam","year":"1975","unstructured":"Temam, R.: On the Euler equations of incompressible perfect fluids. J. Funct. Anal. 20(1), 32\u201343 (1975)","journal-title":"J. Funct. Anal."},{"key":"9175_CR43","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1016\/j.anihpc.2011.05.002","volume":"28","author":"E. Wiedemann","year":"2011","unstructured":"Wiedemann, E.: Existence of weak solutions for the incompressible Euler equations. Ann. Inst. Henri Poincar\u00e9, Anal. Non Lin\u00e9aire 28, 727\u2013730 (2011)","journal-title":"Ann. Inst. Henri Poincar\u00e9, Anal. Non Lin\u00e9aire"},{"key":"9175_CR44","unstructured":"Yahalom, A.: Energy principles for barotropic flows with applications to gaseous disks. Thesis submitted as part of the requirements for the degree of Ph.D. to the Senate of the Hebrew University of Jerusalem (1996)"}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-013-9175-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00332-013-9175-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-013-9175-4","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,7,16]],"date-time":"2019-07-16T01:55:11Z","timestamp":1563242111000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00332-013-9175-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,6,22]]},"references-count":44,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2013,12]]}},"alternative-id":["9175"],"URL":"https:\/\/doi.org\/10.1007\/s00332-013-9175-4","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,6,22]]}}}