1,173 research outputs found
Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications
In this work we study certain invariant measures that can be associated to
the time averaged observation of a broad class of dissipative semigroups via
the notion of a generalized Banach limit. Consider an arbitrary complete
separable metric space which is acted on by any continuous semigroup
. Suppose that possesses a global
attractor . We show that, for any generalized Banach limit
and any distribution of initial
conditions , that there exists an invariant probability measure
, whose support is contained in , such that for all
observables living in a suitable function space of continuous mappings
on .
This work is based on a functional analytic framework simplifying and
generalizing previous works in this direction. In particular our results rely
on the novel use of a general but elementary topological observation, valid in
any metric space, which concerns the growth of continuous functions in the
neighborhood of compact sets. In the case when does not
possess a compact absorbing set, this lemma allows us to sidestep the use of
weak compactness arguments which require the imposition of cumbersome weak
continuity conditions and limits the phase space to the case of a reflexive
Banach space. Two examples of concrete dynamical systems where the semigroup is
known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic
Spectral analysis of semigroups and growth-fragmentation equations
The aim of this paper is twofold: (1) On the one hand, the paper revisits the
spectral analysis of semigroups in a general Banach space setting. It presents
some new and more general versions, and provides comprehensible proofs, of
classical results such as the spectral mapping theorem, some (quantified)
Weyl's Theorems and the Krein-Rutman Theorem. Motivated by evolution PDE
applications, the results apply to a wide and natural class of generators which
split as a dissipative part plus a more regular part, without assuming any
symmetric structure on the operators nor Hilbert structure on the space, and
give some growth estimates and spectral gap estimates for the associated
semigroup. The approach relies on some factorization and summation arguments
reminiscent of the Dyson-Phillips series in the spirit of those used in
[87,82,48,81]. (2) On the other hand, we present the semigroup spectral
analysis for three important classes of "growth-fragmentation" equations,
namely the cell division equation, the self-similar fragmentation equation and
the McKendrick-Von Foerster age structured population equation. By showing that
these models lie in the class of equations for which our general semigroup
analysis theory applies, we prove the exponential rate of convergence of the
solutions to the associated remarkable profile for a very large and natural
class of fragmentation rates. Our results generalize similar estimates obtained
in \cite{MR2114128,MR2536450} for the cell division model with (almost)
constant total fragmentation rate and in \cite{MR2832638,MR2821681} for the
self-similar fragmentation equation and the cell division equation restricted
to smooth and positive fragmentation rate and total fragmentation rate which
does not increase more rapidly than quadratically. It also improves the
convergence results without rate obtained in \cite{MR2162224,MR2114413} which
have been established under similar assumptions to those made in the present
work
Exponential stability for infinite-dimensional non-autonomous port-Hamiltonian Systems
We study the non-autonomous version of an infinite-dimensional
port-Hamiltonian system on an interval . Employing abstract results on
evolution families, we show -well-posedness of the corresponding Cauchy
problem, and thereby existence and uniqueness of classical solutions for
sufficiently regular initial data. Further, we demonstrate that a dissipation
condition in the style of the dissipation condition sufficient for uniform
exponential stability in the autonomous case also leads to a uniform
exponential decay of the energy in this non-autonomous setting
Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions
This paper proves the asymptotic stability of the multidimensional wave
equation posed on a bounded open Lipschitz set, coupled with various classes of
positive-real impedance boundary conditions, chosen for their physical
relevance: time-delayed, standard diffusive (which includes the
Riemann-Liouville fractional integral) and extended diffusive (which includes
the Caputo fractional derivative). The method of proof consists in formulating
an abstract Cauchy problem on an extended state space using a dissipative
realization of the impedance operator, be it finite or infinite-dimensional.
The asymptotic stability of the corresponding strongly continuous semigroup is
then obtained by verifying the sufficient spectral conditions derived by Arendt
and Batty (Trans. Amer. Math. Soc., 306 (1988)) as well as Lyubich and V\~u
(Studia Math., 88 (1988))
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