Wavelets: Tools for Science and Technology by Stéphane Jaffard, Yves Meyer, Robert D. Ryan

By Stéphane Jaffard, Yves Meyer, Robert D. Ryan

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Based on numerical simulations of shell models, we found that there exists a stable fixed point in which the scaling exponents corresponds to the non linear field for a particular choice of the parameter δ. e. independent of δ) and we provided some qualitative arguments to explain the observed universality. At this stage, no claim can be done concerning the relation between the universal fixed point and the scaling of the non linear field. The existence of such relation would imply the universality of the scaling properties of the nonlinear shell models and the Navier-Stokes equations.

Based on numerical simulations of shell models, we found that there exists a stable fixed point in which the scaling exponents corresponds to the non linear field for a particular choice of the parameter δ. e. independent of δ) and we provided some qualitative arguments to explain the observed universality. At this stage, no claim can be done concerning the relation between the universal fixed point and the scaling of the non linear field. The existence of such relation would imply the universality of the scaling properties of the nonlinear shell models and the Navier-Stokes equations.

These results provide a clear answer to the question whether the scaling exponents of the linear model (either shell model or Navier-Stokes) are anomalous. What remains is to find out whether they are also universal, and this is what we attempt to find in the rest of this paper. 5 Fixed Point Properties For the shell model, we consider the chain of equations (s) i dwn = Φn (w(s−1) , w(s) ) − νkn2 wn(s) + fn(s) . dt 3 (12) 42 I. Procaccia et al. Such a chain of equations has, in addition to the dependence on the initial field, also free parameters δ (s) .

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