# Small scale creation for solutions of the incompressible two-dimensional Euler equation

### Abstract

We construct an initial data for the two-dimensional Euler equation in a disk for which the gradient of vorticity exhibits double exponential growth in time for all times. This estimate is known to be sharp — the double exponential growth is the fastest possible growth rate.

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• [Wolibner] W. Wolibner, "Un theorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long," Math. Z., vol. 37, iss. 1, pp. 698-726, 1933.
@article{Wolibner,
author = {Wolibner, W.},
journal = {Math. Z.},
number = {1},
pages = {698--726},
title = {Un theorème sur l'existence du mouvement plan d'un fluide parfait, homogène, incompressible, pendant un temps infiniment long},
volume = {37},
year = {1933},
doi = {10.1007/BF01474610},
issn = {0025-5874},
}
• [Holder] E. Hölder, "Über die unbeschränkte Fortsetzbarkeit einer stetigen ebenen Bewegung in einer unbegrenzten inkompressiblen Flüssigkeit," Math. Z., vol. 37, iss. 1, pp. 727-738, 1933.
@article{Holder,
author = {H{ö}lder, Ernst},
journal = {Math. Z.},
number = {1},
pages = {727--738},
title = {Über die unbeschränkte {F}ortsetzbarkeit einer stetigen ebenen {B}ewegung in einer unbegrenzten inkompressiblen {F}lüssigkeit},
volume = {37},
year = {1933},
doi = {10.1007/BF01474611},
issn = {0025-5874},
}
• [Kato2] T. Kato, "On classical solutions of the two-dimensional non-stationary Euler equation," Arch. Rational Mech. Anal., vol. 25, pp. 188-200, 1967.
@article{Kato2,
author = {Kato, Tosio},
journal = {Arch. Rational Mech. Anal.},
pages = {188--200},
title = {On classical solutions of the two-dimensional non-stationary {E}uler equation},
volume = {25},
year = {1967},
doi = {10.1007/BF00251588},
issn = {0003-9527},
}
• [Koch] H. Koch, "Transport and instability for perfect fluids," Math. Ann., vol. 323, iss. 3, pp. 491-523, 2002.
@article{Koch,
author = {Koch, Herbert},
journal = {Math. Ann.},
number = {3},
pages = {491--523},
title = {Transport and instability for perfect fluids},
volume = {323},
year = {2002},
doi = {10.1007/s002080200312},
issn = {0025-5831},
}
• [MP] C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, New York: Springer-Verlag, 1994, vol. 96.
author = {Marchioro, Carlo and Pulvirenti, Mario},
pages = {xii+283},
publisher = {Springer-Verlag},
series = {Appl. Math. Sci.},
title = {Mathematical Theory of Incompressible Nonviscous Fluids},
volume = {96},
year = {1994},
doi = {10.1007/978-1-4612-4284-0},
isbn = {0-387-94044-8},
}
• [Chemin] J. Chemin, Perfect Incompressible Fluids, New York: The Clarendon Press, Oxford University Press, 1998, vol. 14.
author = {Chemin, Jean-Yves},
pages = {x+187},
publisher = {The Clarendon Press, Oxford University Press},
series = {Oxford Lecture Ser. Math. Appl.},
title = {Perfect Incompressible Fluids},
volume = {14},
year = {1998},
isbn = {0-19-850397-0},
}
• [YudDE] V. I. Yudovich, "The flow of a perfect, incompressible liquid through a given region," Soviet Phys. Dokl., vol. 7, pp. 789-791, 1962.
@article{YudDE,
author = {Yudovich, V. I.},
journal = {Soviet Phys. Dokl.},
pages = {789--791},
title = {The flow of a perfect, incompressible liquid through a given region},
volume = {7},
year = {1962},
issn = {0038-5689},
}
• [Jud1] V. I. Judovivc, "The loss of smoothness of the solutions of Euler equations with time," Dinamika Sploshn. Sredy, vol. 16, pp. 71-78, 121, 1974.
@article{Jud1,
author = {Judovi{\v{c}},
V. I.},
journal = {Dinamika Sploshn. Sredy},
pages = {71--78, 121},
title = {The loss of smoothness of the solutions of {E}uler equations with time},
volume = {16},
year = {1974},
}
• [Yud2] V. I. Yudovich, "On the loss of smoothness of the solutions of the Euler equations and the inherent instability of flows of an ideal fluid," Chaos, vol. 10, iss. 3, pp. 705-719, 2000.
@article{Yud2,
author = {Yudovich, V. I.},
journal = {Chaos},
number = {3},
pages = {705--719},
title = {On the loss of smoothness of the solutions of the {E}uler equations and the inherent instability of flows of an ideal fluid},
volume = {10},
year = {2000},
doi = {10.1063/1.1287066},
issn = {1054-1500},
}
• [MSY] A. Morgulis, A. Shnirelman, and V. Yudovich, "Loss of smoothness and inherent instability of 2D inviscid fluid flows," Comm. Partial Differential Equations, vol. 33, iss. 4-6, pp. 943-968, 2008.
@article{MSY,
author = {Morgulis, Andrey and Shnirelman, Alexander and Yudovich, Victor},
journal = {Comm. Partial Differential Equations},
number = {4-6},
pages = {943--968},
title = {Loss of smoothness and inherent instability of 2{D} inviscid fluid flows},
volume = {33},
year = {2008},
doi = {10.1080/03605300802108016},
issn = {0360-5302},
}
• [Nad1] N. S. Nadirashvili, "Wandering solutions of the two-dimensional Euler equation," Funktsional. Anal. i Prilozhen., vol. 25, iss. 3, pp. 70-71, 1991.
journal = {Funktsional. Anal. i Prilozhen.},
note = {translation in \emph{Funct. Anal. Appl.} {\bf 25} (1992), 220--221},
number = {3},
pages = {70--71},
title = {Wandering solutions of the two-dimensional {E}uler equation},
volume = {25},
year = {1991},
doi = {10.1007/BF01085491},
issn = {0374-1990},
}
• [BC] H. Bahouri and J. -Y. Chemin, "Equations de transport relatives á des champs de vecteurs non-lipschitziens et mécanique des fluides," Arch. Rational Mech. Anal., vol. 127, iss. 2, pp. 159-181, 1994.
@article{BC,
author = {Bahouri, H. and Chemin, J.-Y.},
journal = {Arch. Rational Mech. Anal.},
number = {2},
pages = {159--181},
title = {Equations de transport relatives á des champs de vecteurs non-lipschitziens et mécanique des fluides},
volume = {127},
year = {1994},
doi = {10.1007/BF00377659},
issn = {0003-9527},
}
• [Den1] S. A. Denisov, "Infinite superlinear growth of the gradient for the two-dimensional Euler equation," Discrete Contin. Dyn. Syst., vol. 23, iss. 3, pp. 755-764, 2009.
@article{Den1,
author = {Denisov, Sergey A.},
journal = {Discrete Contin. Dyn. Syst.},
number = {3},
pages = {755--764},
title = {Infinite superlinear growth of the gradient for the two-dimensional {E}uler equation},
volume = {23},
year = {2009},
doi = {10.3934/dcds.2009.23.755},
issn = {1078-0947},
}
• [Den2] S. A. Denisov, Double exponential growth of the vorticity gradient for the two-dimensional Euler equation.
@misc{Den2,
author = {Denisov, Sergey A.},
note = {to appear in \emph{Proc. Amer. Math. Soc.}},
title = {Double exponential growth of the vorticity gradient for the two-dimensional {E}uler equation},
}
• [Den3] S. A. Denisov, The sharp corner formation in 2D Euler dynamics of patches: infinite double exponential rate of merging.
@misc{Den3,
author = {Denisov, Sergey A.},
title = {The sharp corner formation in 2{D} {E}uler dynamics of patches: infinite double exponential rate of merging},
}
• [Tao] T. Tao, Why global regularity for Navier-Stokes is hard.
@misc{Tao,
author = {Tao, T.},
note = {post by {N}ets {K}atz on 20 March, 2007 at 12:21 am and the following thread},
title = {Why global regularity for {N}avier-{S}tokes is hard},
url = {http://terrytao.wordpress.com/2007/03/18/ why-global-regularity-for-navier-stokes-is-hard/},
}
• [Kato] T. Kato, "Remarks on the Euler and Navier-Stokes equations in ${\bf R}^2$," in Nonlinear Functional Analysis and its Applications, Part 2, Providence, RI: Amer. Math. Soc., 1986, vol. 45, pp. 1-7.
author = {Kato, Tosio},
booktitle = {Nonlinear Functional Analysis and its Applications, {P}art 2},
pages = {1--7},
publisher = {Amer. Math. Soc.},
series = {Proc. Sympos. Pure Math.},
title = {Remarks on the {E}uler and {N}avier-{S}tokes equations in {${\bf R}\sp 2$}},
volume = {45},
year = {1986},
doi = {10.1090/pspum/045.2},
}
• [BKM] J. T. Beale, T. Kato, and A. Majda, "Remarks on the breakdown of smooth solutions for the $3$-D Euler equations," Comm. Math. Phys., vol. 94, iss. 1, pp. 61-66, 1984.
@article{BKM,
author = {Beale, J. T. and Kato, T. and Majda, A.},
journal = {Comm. Math. Phys.},
number = {1},
pages = {61--66},
title = {Remarks on the breakdown of smooth solutions for the {$3$}-{D} {E}uler equations},
volume = {94},
year = {1984},
doi = {10.1007/BF01212349},
issn = {0010-3616},
}
• [MB] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Univ. Press, Cambridge, 2002, vol. 27.
@book{MB,
author = {Majda, Andrew J. and Bertozzi, Andrea L.},
pages = {xii+545},
publisher = {Cambridge Univ. Press, Cambridge},
series = {Cambridge Texts Appl. Math.},
title = {Vorticity and Incompressible Flow},
volume = {27},
year = {2002},
isbn = {0-521-63057-6; 0-521-63948-4},
}
• [Evans] L. C. Evans, Partial Differential Equations, Amer. Math. Soc., 1998, vol. 19.
@book{Evans,
author = {Evans, Lawrence C.},
pages = {xviii+662},
publisher = {Amer. Math. Soc.},
title = {Partial Differential Equations},
volume = {19},
year = {1998},
isbn = {0-8218-0772-2},
}
• [GT] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, New York: Springer-Verlag, 2001.
author = {Gilbarg, David and Trudinger, Neil S.},
pages = {xiv+517},
publisher = {Springer-Verlag},
series = {Classics in Math.},
title = {Elliptic Partial Differential Equations of Second Order},
year = {2001},
isbn = {3-540-41160-7},
}
• [Miller1990] J. Miller, "Statistical mechanics of Euler equations in two dimensions," Phys. Rev. Lett., vol. 65, iss. 17, pp. 2137-2140, 1990.
@article{Miller1990,
author = {Miller, Jonathan},
journal = {Phys. Rev. Lett.},
number = {17},
pages = {2137--2140},
title = {Statistical mechanics of {E}uler equations in two dimensions},
volume = {65},
year = {1990},
doi = {10.1103/PhysRevLett.65.2137},
issn = {0031-9007},
}
• [Robert1991] R. Robert, "A maximum-entropy principle for two-dimensional perfect fluid dynamics," J. Statist. Phys., vol. 65, iss. 3-4, pp. 531-553, 1991.
@article{Robert1991,
author = {Robert, Raoul},
journal = {J. Statist. Phys.},
number = {3-4},
pages = {531--553},
title = {A maximum-entropy principle for two-dimensional perfect fluid dynamics},
volume = {65},
year = {1991},
doi = {10.1007/BF01053743},
issn = {0022-4715},
}
• [Shnirelman1993] A. I. Shnirelman, "Lattice theory and flows of ideal incompressible fluid," Russian J. Math. Phys., vol. 1, iss. 1, pp. 105-114, 1993.
@article{Shnirelman1993,
author = {Shnirelman, Alexander I.},
journal = {Russian J. Math. Phys.},
number = {1},
pages = {105--114},
title = {Lattice theory and flows of ideal incompressible fluid},
volume = {1},
year = {1993},
issn = {1061-9208},
}
• [MW2006] A. J. Majda and X. Wang, Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows, Cambridge: Cambridge Univ. Press, 2006.
author = {Majda, Andrew J. and Wang, Xiaoming},
pages = {xii+551},
publisher = {Cambridge Univ. Press},
title = {Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows},
year = {2006},
doi = {10.1017/CBO9780511616778},
isbn = {978-0-521-83441-4; 0-521-83441-4},
}
• [HouLuo] T. Hou and G. Luo, Potentially singular solutions of the 3D incompressible Euler equations.
@misc{HouLuo,
author = {Hou, T. and Luo, G.},
title = {Potentially singular solutions of the 3{D} incompressible {E}uler equations},
}

## Authors

Alexander Kiselev

Rice University, Houston, TX