Communications in Mathematical Physics

, Volume 276, Issue 1, pp 189–220 | Cite as

Simple Systems with Anomalous Dissipation and Energy Cascade

  • Jonathan C. Mattingly
  • Toufic Suidan
  • Eric Vanden-EijndenEmail author


We analyze a class of dynamical systems of the type \(\dot a_n(t) = c_{n-1} a_{n-1}(t) - c_n a_{n+1}(t) + f_n(t), n \epsilon {{\mathbb{N}}}, a_0=0,\) where f n (t) is a forcing term with \(f_n(t)\not = 0\) only for \(n\le n_\star < \infty\) and the coupling coefficients c n satisfy a condition ensuring the formal conservation of energy \(\frac12\sum_n |a_n(t)|^2\) . Despite being formally conservative, we show that these dynamical systems support dissipative solutions (suitably defined) and, as a result, may admit unique (statistical) steady states when the forcing term f n (t) is nonzero. This claim is demonstrated via the complete characterization of the solutions of the system above for specific choices of the coupling coefficients c n . The mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes with higher n; this is responsible for solutions with interesting energy spectra, namely \(\mathbb{E} |a_n|^2\) scales as \(n^{-\alpha}\) as n→∞. Here the exponents α depend on the coupling coefficients c n and \(\mathbb{E}\) denotes expectation with respect to the equilibrium measure. This is reminiscent of the conjectured properties of the solutions of the Navier-Stokes equations in the inviscid limit and their accepted relationship with fully developed turbulence. Hence, these simple models illustrate some of the heuristic ideas that have been advanced to characterize turbulence, similar in that respect to the random passive scalar or random Burgers equation, but even simpler and fully solvable.


Unit Circle Dissipation Rate Simple System Taylor Series Expansion Open Unit Disk 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. CFP06a.
    Cheskidov, A., Friedlander, S., Pavlović, N.: An Inviscid Dyadic Model of Turbulence: The Global Attractor. arXiv:math.AP/06108115 v1, 2006Google Scholar
  2. CFP06b.
    Cheskidov, A., Friedlander, S., Pavlović, N.: An Inviscid Dyadic Model of Turbulence: The Fixed Point and Onsager’s Conjecture., 2006
  3. CET94.
    Constantin P., E W. and Titi E.S. (1994). Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys. 165(1): 207–209 zbMATHCrossRefADSMathSciNetGoogle Scholar
  4. CR06.
    Constantin, P., Ramos, F.: Inviscid limit for damped and driven incompressible Navier-Stokes equations in \({{\mathbb R}^2}\) . v1, 2006
  5. DR00.
    Duchon J. and Robert R. (2000). Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations. Nonlinearity 13(1): 249–255 zbMATHCrossRefADSMathSciNetGoogle Scholar
  6. E01.
    E, W.: Stochastic hydrodynamics. In: Current developments in mathematics, 2000, Somerville, MA: Int. Press, 2001, pp. 109–147Google Scholar
  7. Eyi01.
    Eyink G.L. (2001). Dissipation in turbulent solutions of 2D Euler equations. Nonlinearity 14(4): 787–802 zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. Fel54.
    Feller W. (1954). The general diffusion operator and positivity preserving semi-groups in one dimension. Ann. of Math. (2) 60: 417–436 CrossRefMathSciNetGoogle Scholar
  9. FGV01.
    Falkovich G., Gawȩdzki K. and Vergassola M. (2001). Particles and fields in fluid turbulence. Rev. Mod. Phys. 73(4): 913–975 CrossRefADSGoogle Scholar
  10. FO90.
    Flajolet P. and Odlyzko A. (1990). Singularity analysis of generating functions. SIAM J. Discrete Math. 3(2): 216–240 zbMATHCrossRefMathSciNetGoogle Scholar
  11. Fri95.
    Frisch, U.: Turbulence. Cambridge: Cambridge University Press, 1995Google Scholar
  12. Hil01.
    Hilberdink T. (2001). A Tauberian theorem for power series. Arch. Math. (Basel) 77(4): 354–359 zbMATHMathSciNetGoogle Scholar
  13. McK56.
    McKean H.P. (1956). Elementary solutions for certain parabolic partial differential equations. Trans. Amer. Math. Soc. 82: 519–548 zbMATHCrossRefMathSciNetGoogle Scholar
  14. Sri05.
    Srinivasan, R.: Simple models with cascade of energy and anomalous dissipation. In: Oliver Buhler, Charles Doering, ed. Fast times and fine scales, Woods Hole Oceanographic Institution Technical Reports. Woods Hole Oceanographic Institution, 2005Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Jonathan C. Mattingly
    • 1
  • Toufic Suidan
    • 2
  • Eric Vanden-Eijnden
    • 3
    Email author
  1. 1.Department of Mathematics and CNCSDuke UniversityDurhamUSA
  2. 2.Mathematics DepartmentUniversity of CaliforniaSanta CruzUSA
  3. 3.Courant InstituteNew York UniversityNew YorkUSA

Personalised recommendations