15 research outputs found
Statistical Properties of Nonlinear Shell Models of Turbulence from Linear Advection Models: Rigorous Results
In a recent paper it was proposed that for some nonlinear shell models of
turbulence one can construct a linear advection model for an auxiliary field
such that the scaling exponents of all the structure functions of the linear
and nonlinear fields coincide.
The argument depended on an assumption of continuity of the solutions as a
function of a parameter. The aim of this paper is to provide a rigorous proof
for the validity of the assumption. In addition we clarify here when the swap
of a nonlinear model by a linear one will not work.Comment: 7 pages, 4 figures, submitted to Nonlinearit
Analytic Study of Shell Models of Turbulence
In this paper we study analytically the viscous `sabra' shell model of energy
turbulent cascade. We prove the global regularity of solutions and show that
the shell model has finitely many asymptotic degrees of freedom, specifically:
a finite dimensional global attractor and globally invariant inertial
manifolds. Moreover, we establish the existence of exponentially decaying
energy dissipation range for the sufficiently smooth forcing
A Note on the Regularity of Inviscid Shell Model of Turbulence
In this paper we continue the analytical study of the sabra shell model of
energy turbulent cascade initiated in \cite{CLT05}. We prove the global
existence of weak solutions of the inviscid sabra shell model, and show that
these solutions are unique for some short interval of time. In addition, we
prove that the solutions conserve the energy, provided that the components of
the solution satisfy , for
some positive absolute constant , which is the analogue of the Onsager's
conjecture for the Euler's equations. Moreover, we give a Beal-Kato-Majda type
criterion for the blow-up of solutions of the inviscid sabra shell model and
show the global regularity of the solutions in the ``two-dimensional''
parameters regime
Sharp Lower Bounds for the Dimension of the Global Attractor of the Sabra Shell Model of Turbulence
In this work we derive a 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 and for sufficiently small viscosity term , the Sabra shell model
has a global attractor of large Hausdorff and fractal dimensions proportional
to for all values of the governing parameter
, except for . The obtained lower bounds are sharp,
matching the upper bounds for the dimension of the global attractor obtained in
our previous work. Moreover, we show different scenarios of the transition to
chaos for different parameters regime and for specific forcing. In the
``three-dimensional'' regime of parameters this scenario changes when the
parameter 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-time dynamics of the model becomes trivial for any value of the
viscosity