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### Open Access Journals

DCDS-S

We study the uniform global attractor for a general nonautonomous
reaction-diffusion system without uniqueness using a new developed
framework of an evolutionary system. We prove the existence and the
structure of a weak uniform (with respect to a symbol space) global
attractor $\mathcal A$. Moreover, if the external force is normal,
we show that this attractor is in fact a strong uniform global
attractor. The existence of a uniform (with respect to the initial
time) global attractor $\mathcal A^0$ also holds in this case, but
its relation to $\mathcal A$ is not yet clear due to the
non-uniqueness feature of the system.

DCDS-B

We give an abstract framework for studying nonautonomous PDEs,
called a generalized evolutionary system. In this setting, we define the notion
of a pullback attractor. Moreover, we show that the pullback attractor, in
the weak sense, must always exist. We then study the structure of these attractors and the existence of a strong pullback attractor. We then apply our
framework to both autonomous and nonautonomous evolutionary systems as
they first appeared in earlier works by Cheskidov, Foias, and Lu. In this con-
text, we compare the pullback attractor to both the global attractor (in the
autonomous case) and the uniform attractor (in the nonautonomous case). Finally, we apply our results to the nonautonomous 3D Navier-Stokes equations
on a periodic domain with a translationally bounded force. We show that the
Leray-Hopf weak solutions form a generalized evolutionary system and must
then have a weak pullback attractor.

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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