Energy dissipation and selfsimilar solutions for an unforced inviscid dyadic model
Authors:
D. Barbato, F. Flandoli and F. Morandin
Journal:
Trans. Amer. Math. Soc. 363 (2011), 19251946
MSC (2010):
Primary 35Q35, 76B03, 35Q30
Published electronically:
November 16, 2010
MathSciNet review:
2746670
Fulltext PDF
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Additional Information
Abstract: A shelltype 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. Selfsimilar decaying positive solutions are introduced and proved to exist and classified. Coalescence and blowup are obtained as a consequence, in the class of arbitrary sign solutions.
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 2.
 R. Benzi, B. Levant, I. Procaccia, E.S. Titi, Statistical properties of nonlinear shell models of turbulence from linear advection model: rigorous results, Nonlinearity 20 (2007), no. 6, 14311441. MR 2327131 (2008m:76065)
 3.
 L. Biferale, Shell Models of Energy Cascade in Turbulence, Annu. Rev. Fluid. Mech. 35 (2003), 441468. MR 1967019 (2004b:76074)
 4.
 A. Cheskidov, Blowup in finite time for the dyadic model of the NavierStokes equations, Trans. Amer. Math. Soc. 360 (2008), no. 10, 51015120. MR 2415066 (2010a:35180)
 5.
 A. Cheskidov, S. Friedlander, N. Pavlovic, Inviscid dyadic model of turbulence: The fixed point and Onsager's conjecture, J. Math. Phys. 48 (2007), no. 6, 065503, 16 pp. MR 2337019 (2008k:76012)
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 A. Cheskidov, S. Friedlander, N. Pavlovic, An inviscid dyadic model of turbulence: The global attractor, arXiv:math.AP/0610815.
 7.
 P. Constantin, B. Levant, E. S. Titi, Regularity of inviscid shell models of turbulence, Phys. Review E (3) 75 (2007), no. 1, 016304, 110. MR 2324676 (2008e:76085)
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 U. Frisch, Turbulence, Cambridge University Press, Cambridge (1995). MR 1428905 (98e:76002)
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 N. H. Katz, N. Pavlovic, Finite time blowup for a dyadic model of the Euler equations, Trans. Amer. Math. Soc. 357 (2005), no. 2, 695708. MR 2095627 (2005h:35284)
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 A. Kiselev, A. Zlatoš, On discrete models of the Euler equation, IMRN 38 (2005), no. 38, 23152339. MR 2180809 (2007e:35229)
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 A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluids at very large Reynolds numbers, Dokl. Akad. Nauk. SSSR 30 (1941), 301305. MR 0004146 (2:327d)
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 F. Waleffe, On some dyadic models of the Euler equations, Proc. Amer. Math. Soc. 134 (2006), 29132922. MR 2231615 (2007g:35222)
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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/S000299472010053024
Received by editor(s):
October 30, 2008
Published electronically:
November 16, 2010
Article copyright:
© Copyright 2010
American Mathematical Society
