In a 1997 paper, Ball defined a generalised semiflow as a means to consider the solutions of equations without (or not known to possess) the property of uniqueness. In particular he used this to show that the 3D Navier–Stokes equations have a global attractor provided that all weak solutions are continuous from (0, ∞) into L2. In this paper we...
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June 23, 2015 (v1)PublicationUploaded on: December 4, 2022
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April 8, 2015 (v1)Publication
Some results concerning the stability and stabilisation of stochastic linear partial differential equations in the sense of Stratonovich are proved. The main result ensures that a deterministic linear PDE can be stabilised by adding a suitable Stratonovich noise if and only if the linear partial di erential operator has negative trace.
Uploaded on: March 27, 2023 -
October 21, 2016 (v1)Publication
Two tracking properties for trajectories on attracting sets are studied. We prove that trajectories on the full phase space can be followed arbitrarily closely by skipping from one solution on the global attractor to another. A sufficient condition for asymptotic completeness of invariant exponential attractors is found, obtaining similar...
Uploaded on: March 27, 2023 -
June 23, 2015 (v1)Publication
We consider the three-dimensional Navier–Stokes equations on a periodic domain. We give a simple proof of the local existence of solutions in View the MathML source, and show that the existence of a regular solution on a bounded time interval [0,T] is stable with respect to perturbations of the initial data in View the MathML source and...
Uploaded on: March 27, 2023 -
April 8, 2015 (v1)Publication
This paper presents a comparison between two abstract frameworks in which one can treat multi-valued semiflows and their asymptotic behaviour. We compare the theory developed by Ball [5] to treat equations whose solutions may not be unique, and that due to Melnik & Valero [25] tailored more for differential inclusions. Although they deal with...
Uploaded on: March 27, 2023 -
April 21, 2016 (v1)Publication
Lotka-Volterra systems have been extensively studied by many authors, both in the autonomous and non-autonomous cases. In previous papers the time asymptotic behaviour as t → ∞ has been considered. In this paper we also consider the "pullback" asymptotic behaviour which roughly corresponds to observing a system "now" that has already been...
Uploaded on: March 27, 2023 -
April 20, 2016 (v1)Publication
The goal of this work is to study in some detail the asymptotic behaviour of a non-autonomous Lotka-Volterra model, both in the conventional sense (as t → ∞) and in the "pullback" sense (starting a fixed initial condition further and further back in time). The non-autonomous terms in our model are chosen such that one species will eventually...
Uploaded on: March 27, 2023 -
April 20, 2016 (v1)Publication
In a previous paper we introduced various definitions of stability and instability for non-autonomous differential equations, and applied these to investigate the bifurcations in some simple models. In this paper we present a more systematic theory of local bifurcations in scalar non-autonomous equations.
Uploaded on: March 27, 2023 -
April 21, 2016 (v1)Publication
There is a vast body of literature devoted to the study of bifurcation phenomena in autonomous systems of differential equations. However, there is currently no well-developed theory that treats similar questions for the nonautonomous case. Inspired in part by the theory of pullback attractors, we discuss generalisations of various autonomous...
Uploaded on: March 27, 2023 -
April 8, 2015 (v1)Publication
The relationship between random attractors and global attractors for dynamical systems is studied. If a partial differential equation is perturbed by an ²¡small random term and certain hypotheses are satisfied, the upper semicontinuity of the random attractors is obtained as ² goes to zero. The results are applied to the Navier-Stokes equations...
Uploaded on: December 4, 2022 -
April 8, 2015 (v1)Publication
Using the relatively new concept of a pullback attractor, we present some results on the existence of attractors for differential equations with variable delay. We give a variety of examples to which our result applies.
Uploaded on: March 27, 2023 -
April 8, 2015 (v1)Publication
We study in some detail the structure of the random attractor for the Chafee{Infante reaction{di¬usion equation perturbed by a multiplicative white noise, du = (¢u + u ¡ u3) dt + ¼ u ¯ dWt; x 2 D » Rm: First we prove, for m 65, a lower bound on the dimension of the random attractor, which is of the same order in as the upper bound we...
Uploaded on: March 27, 2023 -
July 6, 2016 (v1)Publication
In this paper we extend the well-known bifurcation theory for autonomous logistic equations to the non-autonomous equation ut − ∆u = λu − b(t)u 2 with b(t) ∈ [b0, B0], 0 < b0 < B0 < 2b0. In particular, we prove the existence of a unique uniformly bounded trajectory that bifurcates from zero as λ passes through the first eigenvalue of the...
Uploaded on: December 4, 2022 -
April 8, 2015 (v1)Publication
We study the asymptotic behaviour of a reaction-diffusion equation, and prove that the addition of multiplicative white noise (in the sense of Itˆo) stabilizes the stationary solution x 0. We show in addition that this stochastic equation has a finite-dimensional random attractor, and from our results conjecture a possible bifurcation scenario.
Uploaded on: December 4, 2022 -
September 14, 2016 (v1)Publication
We provide bounds on the upper box-counting dimension of negatively invariant subsets of Banach spaces, a problem that is easily reduced to covering the image of the unit ball under a linear map by a collection of balls of smaller radius. As an application of the abstract theory we show that the global attractors of a very broad class of...
Uploaded on: December 4, 2022 -
February 26, 2015 (v1)Publication
We investigate the effect of perturbing the Chafee-Infante scalar reaction diffusion equation, ut−Δu = βu−u3, by noise. While a single multiplicative Itˆo noise of sufficient intensity will stabilise the origin, its Stratonovich counterpart leaves the dimension of the attractor essentially unchanged. We then show that a collection of...
Uploaded on: December 4, 2022 -
October 5, 2016 (v1)Publication
We study the stability of attractors under non-autonomous perturbations that are uniformly small in time. While in general the pullback attractors for the nonautonomous problems converge towards the autonomous attractor only in the Hausdorff semi-distance (upper semicontinuity), the assumption that the autonomous attractor has a 'gradient-like'...
Uploaded on: March 27, 2023 -
July 6, 2016 (v1)Publication
In this paper we determine the exact structure of the pullback attractors in non-autonomous problems that are perturbations of autonomous gradient systems with attractors that are the union of the unstable manifolds of a finite set of hyperbolic equilibria. We show that the pullback attractors of the perturbed systems inherit this structure,...
Uploaded on: March 27, 2023 -
May 16, 2016 (v1)Publication
Lotka-Volterra systems are the canonical ecological models used to analyze population dynamics of competition, symbiosis or prey-predator behaviour involving different interacting species in a fixed habitat. Much of the work on these models has been within the framework of infinite-dimensional dynamical systems, but this has frequently been...
Uploaded on: March 27, 2023 -
June 23, 2015 (v1)Publication
This paper treats the existence of pullback attractors for the non-autonomous 2D Navier--Stokes equations in two different spaces, namely L^2 and H^1. The non-autonomous forcing term is taken in L^2_{\rm loc}(\mathbb R;H^{-1}) and L^2_{\rm loc}(\mathbb R;L^2) respectively for these two results: even in the autonomous case it is not...
Uploaded on: March 27, 2023 -
April 20, 2016 (v1)Publication
The goal of this work is to study the forward dynamics of positive solutions for the nonautonomous logistic equation ut − ∆u = λu − b(t)up, with p > 1, b(t) > 0, for all t ∈ R, limt→∞ b(t) = 0. While the pullback asymptotic behaviour for this equation is now well understood, several different possibilities are realised in the forward asymptotic regime.
Uploaded on: December 5, 2022