Communications in Mathematical Physics

, Volume 293, Issue 2, pp 519–543 | Cite as

Entire Solutions of Hydrodynamical Equations with Exponential Dissipation

  • Claude Bardos
  • Uriel Frisch
  • Walter Pauls
  • Samriddhi Sankar Ray
  • Edriss S. Titi
Open Access


We consider a modification of the three-dimensional Navier–Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as \({{{\rm e}^{|k|/k_{\rm d}}}}\) at high wavenumbers |k|. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than \({{{\rm e}^{-C(k/k_{\rm d})\,{\rm ln}(|k|/k_{\rm d})}}}\) for any C < 1/(2 ln 2). The same result holds for the one-dimensional Burgers equation with exponential dissipation but can be improved: heuristic arguments and very precise simulations, analyzed by the method of asymptotic extrapolation of van der Hoeven, indicate that the leading-order asymptotics is precisely of the above form with C = C * = 1/ ln 2. The same behavior with a universal constant C * is conjectured for the Navier–Stokes equations with exponential dissipation in any space dimension. This universality prevents the strong growth of intermittency in the far dissipation range which is obtained for ordinary Navier–Stokes turbulence. Possible applications to improved spectral simulations are briefly discussed.


Entire Function Burger Equation Hydrodynamical Equation High Wavenumbers Entire Solution 
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We thank J.-Z. Zhu and A. Wirth for important input and M. Blank, K. Khanin, B. Khesin and V. Zheligovsky for many remarks. CB acknowledges the warm hospitality of the Weizmann Institute and SSR that of the Observatoire de la Côte d’Azur, places where parts of this work were carried out. The work of EST was supported in part by the NSF grant No. DMS-0708832 and the ISF grant No. 120/06. SSR thanks R. Pandit, D. Mitra and P. Perlekar for useful discussions and acknowledges DST and UGC (India) for support and SERC (IISc) for computational resources. UF, WP and SSR were partially supported by ANR “OTARIE” BLAN07-2_183172 and used the Mésocentre de calcul of the Observatoire de la Côte d’Azur for computations.

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Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Claude Bardos
    • 1
  • Uriel Frisch
    • 2
  • Walter Pauls
    • 3
  • Samriddhi Sankar Ray
    • 4
  • Edriss S. Titi
    • 5
    • 6
  1. 1.Université Denis Diderot and Laboratoire J.L. Lions, Université Pierre et Marie CurieParisFrance
  2. 2.UNS, CNRS, Laboratoire Cassiopée, OCANice cedex 4France
  3. 3.Max Planck Institute for Dynamics and Self-OrganizationGöttingenGermany
  4. 4.Center for Condensed Matter Theory, Department of PhysicsIndian Institute of ScienceBangaloreIndia
  5. 5.Department of Mathematics and Department of Mechanical and Aerospace EngineeringUniversity of IrvineIrvineUSA
  6. 6.Department of Computer Science and Applied MathematicsWeizmann Institute of ScienceRehovotIsrael

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