The Kneser theorem for ordinary differential equations without uniqueness says
that the attainability set is compact and connected at each instant of time.
We establish corresponding results for the attainability set of weak solutions
for the 3D Navier-Stokes equations satisfying an energy inequality. First, we
present a simplified proof of our earlier result with respect to the weak
topology in the space $H$. Then we prove that this result also holds with
respect to the strong topology on $H$ provided that the weak solutions
satisfying the weak version of the energy inequality are continuous. Finally,
using these results, we show the connectedness of the global attractor of a
family of setvalued semiflows generated by the weak solutions of the NSE
satisfying suitable properties.
Converse Lyapunov theorems are presented for nonautonomous systems
modelled as skew product flows. These characterize various types
of stability of invariant sets and pullback, forward and uniform
attractors in such nonautonomous systems.
A one-step numerical scheme with variable
time--steps is applied to an autonomous differential equation with a uniformly
asymptotically stable set, which is compact but otherwise of arbitrary
geometric shape. A Lyapunov function characterizing this set is used to
show that the
resulting nonautonomous difference equation generated by the numerical scheme
has an absorbing set. The existence of a cocycle attractor consisting of a
of equivariant sets for the associated discrete time cocycle is then
established and shown to be close in the Hausdorff separation to the original
stable set for sufficiently small maximal time-steps.
The $\omega$-limit set $\omega_B$ of a nonautonomous dynamical system generated by a nonautonomous ODE with a positive invariant compact absorbing set $B$ is shown to be asymptotic positive invariant in general and asymptotic negative invariant if, in addition, the vector field is uniformly continuous
in time on the absorbing set. This set has been called the forward attracting set of the nonautonomous dynamical system and is related to Vishik's concept of a uniform attractor.
If $\omega_B$ is also assumed to be uniformly attracting, then its upper semi continuity in a parameter and the upper semi continuous convergence of its counterparts under discretisation by the implicit Euler scheme are established.
Alexei Vadimovich Pokrovskii was an outstanding mathematician, a scientist with very broad mathematical interests, and
a pioneer in the mathematical theory of systems with hysteresis. He died unexpectedly on September 1, 2010 at the age 62.
For the previous nine years he had been Professor and Head of Applied Mathematics at University College Cork in Ireland.
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Inspired by biological phenomena with effects of switching off (maybe just for a while), we investigate non-autonomous reaction-diffusion inclusions whose multi-valued reaction term may depend on the essential supremum over a time interval in the recent past (but) pointwise in space. The focus is on sufficient conditions for the existence of pullback attractors. If the multi-valued reaction term satisfies a form of inclusion principle standard tools for non-autonomous dynamical systems in metric spaces can be applied and provide new results (even) for infinite time intervals of delay. More challenging is the case without assuming such a monotonicity assumption. Then we consider the parabolic differential inclusion with the time interval of delay depending on space and extend the approaches of norm-to-weak semigroups to a purely metric setting. This provides completely new tools for proving pullback attractors of non-autonomous dynamical systems in metric spaces.
The existence of a uniform attractor in a space of higher
regularity is proved for the multi-valued process associated
with the nonautonomous reaction-diffusion equation on an unbounded
domain with delays for which the uniqueness of solutions need not hold. A new method for
checking the asymptotical upper-semicompactness of the solutions is used.
The existence of a random attactor is established for a mean-square random dynamical system (MS-RDS) generated by a stochastic delay equation (SDDE) with random delay for which the drift term is dominated by a nondelay component satisfying a one-sided dissipative Lipschitz condition. It is shown by Razumikhin-type techniques that the solution of this SDDE is ultimately bounded in the mean-square sense and that solutions for different initial values converge exponentially together as time increases in the mean-square sense. Consequently, similar boundedness and convergence properties hold for the MS-RDS and imply the existence of a mean-square random attractor for the MS-RDS that consists of a single stochastic process.
The theory of dynamical systems has undergone some spectacular and
fascinating developments in the past century, as the readers of
this journal are well aware, with the focus predominately on
autonomous systems. There are many ways in which one could classify
the work that has been done, but one that stands clearly in the
forefront is the distinction between dissipative systems with their
attractors and conservative systems, in particular Hamiltonian
Another classification is between autonomous and
nonautonomous systems. Of course, the latter subsumes the former as
special case, but with the former having special structural
features, i.e., the semigroup evolution property, which has allowed
an extensive and seemingly complete theory to be developed. Although
not as extensive, there have also been significant developments in
the past half century on nonautonomous dynamical systems, in
particular the skew-product formalism involving a cocycle evolution
property which generalizes the semigroup property of autonomous
systems. This has been enriched in recent years by advances on
random dynamical systems, which are roughly said a measure theoretic
version of a skew-product flow. In particular, new concepts of
random and nonautonomous attractors have been introduced and
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Discrete-time discrete-state random Markov chains with a tridiagonal
generator are shown to have a random attractor consisting of singleton
subsets, essentially a random path, in the simplex of probability vectors.
The proof uses the Hilbert projection metric and the fact that the linear
cocycle generated by the Markov chain is a uniformly contractive mapping
of the positive cone into itself. The proof does not involve probabilistic
properties of the sample path $\omega$ and is thus equally valid in the
nonautonomous deterministic context of Markov chains with, say,
periodically varying transitions probabilities, in which case the
attractor is a periodic path.