Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Double exponential growth of the vorticity gradient for the two-dimensional Euler equation

Author: Sergey A. Denisov
Journal: Proc. Amer. Math. Soc. 143 (2015), 1199-1210
MSC (2010): Primary 76B99; Secondary 76F99
Published electronically: October 15, 2014
MathSciNet review: 3293735
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For the two-dimensional Euler equation on the torus, we prove that the $ L^\infty $-norm of the vorticity gradient can grow as double exponential over arbitrary long but finite time provided that at time zero it is already sufficiently large. The method is based on the perturbative analysis around the singular stationary solution studied by Bahouri and Chemin in 1994. Our result on the growth of the vorticity gradient is equivalent to the statement that the operator of Euler evolution is not bounded in the linear sense in Lipschitz norm for any time $ t>0$.

References [Enhancements On Off] (What's this?)

  • [1] H. Bahouri and J.-Y. Chemin, Équations de transport relatives á des champs de vecteurs non-lipschitziens et mécanique des fluides, Arch. Rational Mech. Anal. 127 (1994), no. 2, 159-181 (French, with French summary). MR 1288809 (95g:35164),
  • [2] Andrew J. Majda and Andrea L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002. MR 1867882 (2003a:76002)
  • [3] Jean-Yves Chemin, Two-dimensional Euler system and the vortex patches problem, Handbook of mathematical fluid dynamics. Vol. III, North-Holland, Amsterdam, 2004, pp. 83-160. MR 2099034 (2005m:76019)
  • [4] Dongho Chae, Peter Constantin, and Jiahong Wu, Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations, Arch. Ration. Mech. Anal. 202 (2011), no. 1, 35-62. MR 2835862 (2012j:76017),
  • [5] Sergey A. Denisov, Infinite superlinear growth of the gradient for the two-dimensional Euler equation, Discrete Contin. Dyn. Syst. 23 (2009), no. 3, 755-764. MR 2461825 (2010a:35191),
  • [6] Misha Vishik and Susan Friedlander, Nonlinear instability in two dimensional ideal fluids: the case of a dominant eigenvalue, Comm. Math. Phys. 243 (2003), no. 2, 261-273. MR 2021907 (2004k:76055),
  • [7] A. Alexandrou Himonas and Gerard Misiołek, Non-uniform dependence on initial data of solutions to the Euler equations of hydrodynamics, Comm. Math. Phys. 296 (2010), no. 1, 285-301. MR 2606636 (2011c:35429),
  • [8] V. I. Judovič, The loss of smoothness of the solutions of Euler equations with time, Dinamika Splošn. Sredy Vyp. 16 Nestacionarnye Problemy Gidrodinamiki (1974), 71-78, 121 (Russian). MR 0454419 (56 #12670)
  • [9] A. Kiselev and F. Nazarov, A simple energy pump for periodic 2D QGE, in Nonlinear Partial Differential Equations, Proceedings of the Abel Symposium, 2010.
  • [10] Carlo Marchioro and Mario Pulvirenti, Mathematical theory of incompressible nonviscous fluids, Applied Mathematical Sciences, vol. 96, Springer-Verlag, New York, 1994. MR 1245492 (94k:76001)
  • [11] Andrey Morgulis, Alexander Shnirelman, and Victor Yudovich, Loss of smoothness and inherent instability of 2D inviscid fluid flows, Comm. Partial Differential Equations 33 (2008), no. 4-6, 943-968. MR 2424384 (2009h:76014),
  • [12] N. S. Nadirashvili, Wandering solutions of the two-dimensional Euler equation, Funktsional. Anal. i Prilozhen. 25 (1991), no. 3, 70-71 (Russian); English transl., Funct. Anal. Appl. 25 (1991), no. 3, 220-221 (1992). MR 1139875,
  • [13] W. Wolibner, Un theorème sur l'existence du mouvement plan d'un fluide parfait, homogène, incompressible, pendant un temps infiniment long, Math. Z. 37 (1933), no. 1, 698-726 (French). MR 1545430,
  • [14] 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 10 (2000), no. 3, 705-719. MR 1791984 (2002i:76010),

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 76B99, 76F99

Retrieve articles in all journals with MSC (2010): 76B99, 76F99

Additional Information

Sergey A. Denisov
Affiliation: Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, Wisconsin 53706-1388

Keywords: Two-dimensional Euler equation, growth of the vorticity gradient
Received by editor(s): April 4, 2013
Received by editor(s) in revised form: June 5, 2013
Published electronically: October 15, 2014
Additional Notes: This research was supported by NSF grants DMS-1067413, DMS-0758239 and DMS-1159133
Communicated by: Joachim Krieger
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society