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September  2010, 26(3): 781-794. doi: 10.3934/dcds.2010.26.781

An inviscid dyadic model of turbulence: The global attractor

Alexey Cheskidov 1, , Susan Friedlander 2, and  Nataša Pavlović 3,

1. 

Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Offices (M/C 249), 851 S. Morgan Street, Chicago, IL 60607-7045, United States
2. 

Department of Mathematics, University of Southern California, 3620 South Vermont Ave., KAP 108, Los Angeles, CA 90089, United States
3. 

Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200 Austin, Texas 78712

Received  August 2008 Revised  November 2009 Published  December 2009

Properties of an infinite system of nonlinearly coupled ordinary differential equations are discussed. This system models some properties present in the equations of motion for an inviscid fluid such as the skew symmetry and the 3-dimensional scaling of the quadratic nonlinearity. In a companion paper [8] it is proved that every solution for the system with forcing blows up in finite time in the Sobolev $H^{5/6}$ norm. In this present paper, it is proved that after the blow-up time all solutions stay in $H^s$, $s < 5/6$ for almost all time. It is proved that the model system exhibits the phenomenon of anomalous (or turbulent) dissipation which was conjectured for the Euler equations by Onsager. As a consequence of this anomalous dissipation the unique equilibrium of the system is a global attractor.
Keywords: global attractor, turbulence., Dyadic shell model.
Mathematics Subject Classification: Primary: 35Q31; Secondary: 37L3.
Citation: Alexey Cheskidov, Susan Friedlander, Nataša Pavlović. An inviscid dyadic model of turbulence: The global attractor. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 781-794. doi: 10.3934/dcds.2010.26.781
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