Attractors for modulation equations on unbounded domains-existence and comparison

Abstract

We are interested in the long{time behavior of nonlinear parabolic PDEs de ned on unbounded cylindrical domains. For dissipative systems de ned on bounded domains, the long{time behavior can often be described by the dynamics in their nite{dimensional attractors. For systems de ned on the in nite line, very little is known at present, since the lack of compactness prevents application of the standard existence theory for attractors. We develop here an abstract theorem based on the interaction of a uniform and a localizing (weighted) norm which allows us to de ne global attractors for some dissipative problems on unbounded domains such as the Swift{Hohenberg and the Ginzburg{Landau equation. The second aim of this paper is the comparison of attractors. The so{called Ginzburg{Landau formalism allows us to approximate solutions of weakly unstable systems which exhibit modulated periodic patterns. Here we show that the attractor of the Swift{Hohenberg equation is upper semicontinuous in a particular limit to the attractor of the associated Ginzburg{Landau equation.

Freezing Solutions of Equivariant Evolution Equations

“…It is proved in [20] that this is also true for certain cubic nonlinearities for the uniformly local spaces (cf. Example 2.14)…”

“…Another choice is that of locally uniform spaces as proposed in [20], [19]. Take a positive and integrable weight function…”

“…The construction in Proposition 2.12 then yields the locally uniform spaces (using the notation from [20], [19])…”

Spatially Nondecaying Solutions of the 2d Navier-Stokes Equation in a Strip

“…We note however that, in contrast to the dissipative systems in bounded domains, in unbounded ones the global attractor is usually not compact in the initial phase space ( in our case). That is the reason why we need to use the following weaker definition of a global attractor (following [6], [9], [18]). (8.19) where the constant K is independent of c and g. Proof.…”

Stability of Traveling Waves in Partly Parabolic Systems

“…We remark that a less traditional class of function spaces that allow bounded perturbations to fronts is the uniformly local spaces introduced in [18] and studied in detail in [42]. Uniformly local spaces have been used to study stability of fronts that undergo a Turing or Hopf bifurcation in the wake of the front in [8,24]; they allow one to obtain a priori estimates for the periodic perturbations.…”