Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time
Vladimir V. Chepyzhov Mark I. Vishik
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.
keywords: reaction-diffusion systems partly dissipative systems. Trajectory attractor vanishing diffusion
Trajectory attractors for non-autonomous dissipative 2d Euler equations
Vladimir V. Chepyzhov
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.
keywords: zero viscosity limit. dissipative 2d Euler system Trajectory attractor
Alain Miranville Vladimir V. Chepyzhov
-- 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.
A minimal approach to the theory of global attractors
Vladimir V. Chepyzhov Monica Conti Vittorino Pata
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.
keywords: absorbing and attracting sets Semigroups global attractors invariant sets.
Totally dissipative dynamical processes and their uniform global attractors
Vladimir V. Chepyzhov Monica Conti Vittorino Pata
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.
keywords: absorbing and attracting sets Dynamical processes uniform global attractors.
On the convergence of solutions of the Leray-$\alpha $ model to the trajectory attractor of the 3D Navier-Stokes system
Vladimir V. Chepyzhov E. S. Titi Mark I. Vishik
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 $\alpha \rightarrow 0\+.$
keywords: trajectory attractors. 3D Navier--Stokes system 3D Leray-$\alpha $ model

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