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DCDS

We discuss some historical background concerning a modified version of the
Navier-Stokes equations for the motion of an incompressible fluid. The classical
(Newtonian) linear relation between the Cauchy stress tensor and the rate of
strain tensor yields the Navier-Stokes equations. Certain nonlinear relations are
also consistent with basic physical principals and result in equations with
"stronger" dissipation. We describe a class of models that has its genesis
in Kolmogorov's similarity hypothesis for 3-dimensional isotropic turbulence
and was formulated by Smagorinsky in the meteorological context of rapidly
rotating fluids and more generally by Ladyzhenskaya. These models also
describe the motion of fluids with shear dependent viscosities and have received
considerable attention. We present a dyadic model for such modified Navier-Stokes
equations. This model is an example of a hierarchical shell model.
Following the treatment of a (non-physically motivated) linear hyper-dissipative
model given by Katz-Pavlović, we prove for the dyadic model a bound for the
Hausdorff dimension of the singular set at the first time of blow up.
The result interpolates between the results of solvability for sufficiently
strong dissipation of Ladyzhenskaya, (later strengthened by
Nečas et al)
and the bound for the dimension of the singular set for the Navier-Stokes
equations proved by Caffarelli, Kohn and Nirenberg. We discuss the implications
of this dyadic model for the modified Navier-Stokes equation themselves.

DCDS

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.

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