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### Open Access Journals

DCDS

We consider a non-autonomous reaction-diffusion system
of two equations having in one equation a diffusion coefficient
depending on time ($\delta =\delta (t)\geq 0,t\geq 0$) such that
$\delta (t)\rightarrow 0$ as $t\rightarrow +\infty $. The
corresponding Cauchy problem has global weak solutions, however
these solutions are not necessarily unique. We also study the
corresponding "limit'' autonomous system for $\delta =0.$ This
reaction-diffusion system is partly dissipative. We construct the
trajectory attractor A for the limit system. We prove
that global weak solutions of the original non-autonomous system
converge as $t\rightarrow +\infty $ to the set A in a
weak sense. Consequently, A is also as the trajectory
attractor of the original non-autonomous reaction-diffusions
system.

DCDS-B

We construct the trajectory attractor $\mathfrak{A}_{\Sigma }$ for
the non-autonomous dissipative 2d Euler systems with periodic
boundary conditions that contain time dependent dissipation terms
$-r(t)u$ such that $0<\alpha \le r(t)\le \beta$, for $t\ge 0$.
External forces $g(x,t),x\in \mathbb{T}^{2},t\ge 0,$ also depend on
time. The corresponding non-autonomous dissipative 2d
Navier--Stokes systems with the same terms $-r(t)u$ and $g(x,t)$
and with viscosity $\nu >0$ also have the trajectory attractor
$\mathfrak{A}_{\Sigma }^{\nu }.$ Such systems model large-scale
geophysical processes in atmosphere and ocean. We prove that
$\mathfrak{A}_{\Sigma }^{\nu }\rightarrow \mathfrak{A}_{\Sigma }$
as viscosity $\nu \rightarrow 0+$ in the corresponding metric
space. Moreover, we establish the existence of the minimal limit
$\mathfrak{A}_{\Sigma }^{\min }\subseteq \mathfrak{A}_{\Sigma }$ of
the trajectory attractors $\mathfrak{A}_{\Sigma }^{\nu }$ as $\nu
\rightarrow 0+.$ Every set $\mathfrak{A}_{\Sigma }^{\nu }$ is
connected. We prove that $\mathfrak{A}_{\Sigma }^{\min }$ is a
connected invariant subset of $\mathfrak{A}_{\Sigma }.$ The problem
of the connectedness of the trajectory attractor
$\mathfrak{A}_{\Sigma }$ itself remains open.

DCDS

We consider the 3D Navier-Stokes systems with randomly rapidly oscillating right-hand sides. Under the assumption that the random functions are ergodic and statistically homogeneous in space variables or in time variables we prove that the trajectory attractors of these systems tend to the trajectory attractors of homogenized 3D Navier-Stokes systems whose right-hand sides are the average of the corresponding terms of the original systems. We do not assume that the Cauchy problem for the considered 3D Navier-Stokes systems is uniquely solvable.

Bibliography: 44 titles.

CPAA

*-- Mark Iosifovich, how do You think, which scientists have influenced You in the very beginning of Your academic career?*

If viewed chronologically, there were, first of all, my teachers at Lvov State University, which I entered in 1939. The University formerly bore the name of king Kazimir and then became the Ivan Franko University, where the Dean of the Mathematics Department Stefan Banach, a brilliant mathematician, worked. We were taught by the most outstanding professors of the Banach's school: Bronislaw Knaster -- analytical geometry, Yuliush Schauder -- theoretical mechanics, Professor Stanislaw Mazur -- differential geometry. Professor Vladislav Orlicz gave lectures on algebra. All this teaching was in Polish. Only the Deputy Dean Professor Myron Zaritsky gave lectures in Ukrainian.

keywords:

DCDS

For a semigroup $S(t):X\to X$ acting
on a metric space $(X,d)$, we give a notion of global attractor
based only on the minimality with respect to the attraction property.
Such an attractor is shown to be invariant whenever $S(t)$ is

*asymptotically closed*. As a byproduct, we generalize earlier results on the existence of global attractors in the classical sense.
CPAA

We discuss the existence of the global attractor for a family of processes $U_\sigma(t,\tau)$ acting on
a metric space $X$ and depending on a symbol $\sigma$ belonging to some other metric space $\Sigma$.
Such an attractor
is uniform with respect to $\sigma\in\Sigma$, as well as
with respect to the choice of the initial time $\tau\in R$. The existence of the attractor is established
for totally dissipative processes
without any continuity assumption. When the process satisfies some additional (but rather mild)
continuity-like hypotheses, a characterization of the attractor is given.

DCDS-B

Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms

We consider complex Ginzburg-Landau (GL) type equations of the form:

${\partial _t}u = (1 + \alpha i)\Delta u + R{\mkern 1mu} u + (1 + \beta i)|u{|^2}u + g,$ |

where

,

, and

are random rapidly oscillating real functions. Assuming that the random functions are ergodic and statistically homogeneous in space variables, we prove that the trajectory attractors of these systems tend to the trajectory attractors of the homogenized equations whose terms are the average of the corresponding terms of the initial systems.

$R$ |

$β$ |

$g$ |

Bibliography: 52 titles.

DCDS

We study the relations between the global dynamics of the 3D
Leray-$\alpha $ model and the 3D Navier-Stokes system. We prove
that time shifts of bounded sets of solutions of the Leray-$\alpha
$ model converges to the trajectory
attractor of the 3D Navier-Stokes system as time tends to infinity and $
\alpha $ approaches zero. In particular, we show that the
trajectory attractor of the Leray-$\alpha $ model converges to the
trajectory attractor of the 3D Navier-Stokes system when $\alpha
\rightarrow 0\+.$

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