TY - JOUR
T1 - Sharp lower bounds for the dimension of the global attractor of the Sabra shell model of turbulence
AU - Constantin, Peter
AU - Levant, Boris
AU - Titi, Edriss S.
N1 - Funding Information:
The authors would like to thank U. Frisch for a very stimulating discussion, and his valuable comments on the first version of the manuscript. The work of P.C. was supported in part by the NSF grant No. DMS–0504213. The work of E.S.T. was also supported in part by the NSF grant No. DMS–0504619, the MAOF Fellowship of the Israeli Council of Higher Education, the ISF grant No. 120/06, and by the BSF grant No. 2004271.
PY - 2007/6
Y1 - 2007/6
N2 - In this work we derive lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing term and for sufficiently small viscosity term ν, the Sabra shell model has a global attractor of large Hausdorff and fractal dimensions proportional to log∈ ν -1 for all values of the governing parameter ε, except for ε =1. The obtained lower bounds are sharp, matching the upper bounds for the dimension of the global attractor obtained in our previous work. Moreover, the complexity of the dynamics of the shell model increases as the viscosity ν tends to zero, and we describe a precise scenario of successive bifurcations for different parameters regimes. In the "three-dimensional" regime of parameters this scenario changes when the parameter ε becomes sufficiently close to 0 or to 1. We also show that in the "two-dimensional" regime of parameters, for a certain non-zero forcing term, the long-term dynamics of the model becomes trivial for every value of the viscosity.
AB - In this work we derive lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing term and for sufficiently small viscosity term ν, the Sabra shell model has a global attractor of large Hausdorff and fractal dimensions proportional to log∈ ν -1 for all values of the governing parameter ε, except for ε =1. The obtained lower bounds are sharp, matching the upper bounds for the dimension of the global attractor obtained in our previous work. Moreover, the complexity of the dynamics of the shell model increases as the viscosity ν tends to zero, and we describe a precise scenario of successive bifurcations for different parameters regimes. In the "three-dimensional" regime of parameters this scenario changes when the parameter ε becomes sufficiently close to 0 or to 1. We also show that in the "two-dimensional" regime of parameters, for a certain non-zero forcing term, the long-term dynamics of the model becomes trivial for every value of the viscosity.
KW - Dynamic models
KW - Navier-Stokes equations
KW - Shell models
KW - Turbulence
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U2 - 10.1007/s10955-007-9317-x
DO - 10.1007/s10955-007-9317-x
M3 - Article
AN - SCOPUS:34249883569
SN - 0022-4715
VL - 127
SP - 1173
EP - 1192
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
IS - 6
ER -