## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
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- Electronic Research Announcements
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- AIMS Mathematics

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.

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

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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