Energy dissipation and self-similar solutions for an unforced inviscid dyadic model

Authors:
D. Barbato, F. Flandoli and F. Morandin

Journal:
Trans. Amer. Math. Soc. **363** (2011), 1925-1946

MSC (2010):
Primary 35Q35, 76B03, 35Q30

Published electronically:
November 16, 2010

MathSciNet review:
2746670

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Abstract | References | Similar Articles | Additional Information

Abstract: A shell-type model of an inviscid fluid, previously considered in the literature, is investigated in absence of external force. Energy dissipation of positive solutions is proved, and decay of energy like is established. Self-similar decaying positive solutions are introduced and proved to exist and classified. Coalescence and blow-up are obtained as a consequence, in the class of arbitrary sign solutions.

**1.**D. Barbato, M. Barsanti, H. Bessaih, and F. Flandoli,*Some rigorous results on a stochastic GOY model*, J. Stat. Phys.**125**(2006), no. 3, 677–716. MR**2281460**, 10.1007/s10955-006-9203-y**2.**Roberto Benzi, Boris Levant, Itamar Procaccia, and Edriss S. Titi,*Statistical properties of nonlinear shell models of turbulence from linear advection models: rigorous results*, Nonlinearity**20**(2007), no. 6, 1431–1441. MR**2327131**, 10.1088/0951-7715/20/6/006**3.**Luca Biferale,*Shell models of energy cascade in turbulence*, Annual review of fluid mechanics, Vol. 35, Annu. Rev. Fluid Mech., vol. 35, Annual Reviews, Palo Alto, CA, 2003, pp. 441–468. MR**1967019**, 10.1146/annurev.fluid.35.101101.161122**4.**Alexey Cheskidov,*Blow-up in finite time for the dyadic model of the Navier-Stokes equations*, Trans. Amer. Math. Soc.**360**(2008), no. 10, 5101–5120. MR**2415066**, 10.1090/S0002-9947-08-04494-2**5.**Alexey Cheskidov, Susan Friedlander, and Nataša Pavlović,*Inviscid dyadic model of turbulence: the fixed point and Onsager’s conjecture*, J. Math. Phys.**48**(2007), no. 6, 065503, 16. MR**2337019**, 10.1063/1.2395917**6.**A. Cheskidov, S. Friedlander, N. Pavlovic, An inviscid dyadic model of turbulence: The global attractor, arXiv:math.AP/0610815.**7.**Peter Constantin, Boris Levant, and Edriss S. Titi,*Regularity of inviscid shell models of turbulence*, Phys. Rev. E (3)**75**(2007), no. 1, 016304, 10. MR**2324676**, 10.1103/PhysRevE.75.016304**8.**Susan Friedlander and Nataša Pavlović,*Blowup in a three-dimensional vector model for the Euler equations*, Comm. Pure Appl. Math.**57**(2004), no. 6, 705–725. MR**2038114**, 10.1002/cpa.20017**9.**Uriel Frisch,*Turbulence*, Cambridge University Press, Cambridge, 1995. The legacy of A. N. Kolmogorov. MR**1428905****10.**Giovanni Gallavotti,*Foundations of fluid dynamics*, Texts and Monographs in Physics, Springer-Verlag, Berlin, 2002. Translated from the Italian. MR**1872661****11.**Nets Hawk Katz and Nataša Pavlović,*Finite time blow-up for a dyadic model of the Euler equations*, Trans. Amer. Math. Soc.**357**(2005), no. 2, 695–708 (electronic). MR**2095627**, 10.1090/S0002-9947-04-03532-9**12.**Alexander Kiselev and Andrej Zlatoš,*On discrete models of the Euler equation*, Int. Math. Res. Not.**38**(2005), 2315–2339. MR**2180809**, 10.1155/IMRN.2005.2315**13.**A. Kolmogoroff,*The local structure of turbulence in incompressible viscous fluid for very large Reynold’s numbers*, C. R. (Doklady) Acad. Sci. URSS (N.S.)**30**(1941), 301–305. MR**0004146****14.**Fabian Waleffe,*On some dyadic models of the Euler equations*, Proc. Amer. Math. Soc.**134**(2006), no. 10, 2913–2922 (electronic). MR**2231615**, 10.1090/S0002-9939-06-08293-1

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Additional Information

**D. Barbato**

Affiliation:
Dipartimento di Matematica Pura e Applicata, Università degli Studi di Padova, Via Treste 63, 35121 Padova, Italy

**F. Flandoli**

Affiliation:
Department of Applied Mathematics, University of Pisa, Via Buonarroti 1, 56127 Pisa, Italy

**F. Morandin**

Affiliation:
Department of Mathematics, University of Parma, Parco Scienze 53A, 43124 Parma, Italy

DOI:
http://dx.doi.org/10.1090/S0002-9947-2010-05302-4

Received by editor(s):
October 30, 2008

Published electronically:
November 16, 2010

Article copyright:
© Copyright 2010
American Mathematical Society