Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces

Abstract

We derive general existence theorems for random pullback exponential attractors and deduce explicit bounds for their fractal dimension. The results are formulated for asymptotically compact random dynamical systems in Banach spaces.

Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ3

“…, which together with (17) implies e −bz(ϑr−t ω) f (x, e bz(ϑr−t ω) v) ≤ c C e bz(ϑr−t ω) M (ϑ r−t ω) + β e − bz(ϑr−t ω) .…”

“…Proof. Taking the inner product of (61) with y(r), and by e −bz(ϑr−t ω) (f (x, e bz(ϑr−t ω) v (r)) − f (x, e bz(ϑr−t ω) v (r))), y(r) ≤ c y(r) , (by (17)) e −bz(ϑr−t ω) (g(x, σ (r)) −g(x, σ (r))), y(r)…”

“…According to [16], we have a criterion for the existence of a random exponential attractor for a continuous random dynamical system on a separable Hilbert space. Other results on existence criteria of a random exponential attractor for a random dynamical system can be found in [17,18]. With the help of the concept of skew-product cocycle (i.e., random dynamical system) introduced in [15], we consider the existence of the random exponential attractor for a continuous skew-product cocycle (generated by a jointly continuous NRDS ϕ and a base ow θ) on the extended phase space T k ×E, and project this random exponential attractor onto the phase space E. Then we obtain the random uniform exponential attractor for the jointly continuous NRDS ϕ. Consequently, we formulate Theorem 2.8 on the existence of a random uniform exponential attractor for a jointly continuous NRDS.…”

Strong <inline-formula><tex-math id="M1">$ (L^2,L^\gamma\cap H_0^1) $</tex-math></inline-formula>-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension

“…As already noted in Remark 3.5, Theorem 3.4 indicates a smoothing property of the semigroup of (2.1), which is known useful in estimating the upper bounds of the dimensions of a global attractor as well as in constructing an exponential attractor and further estimating its attracting rate, see, e.g., [8,12,4], etc. In the following we study the fractal dimension of the global attractor A and its translation A − z 0 as an example to make use of the new smoothing properties.…”

Dynamics of Stochastic Reaction-Diffusion Equations

“…Naturally, there exist further probabilistic methods that provide dynamical insights, which are based on Kolmogorov/Fokker-Planck equations associated to SPDEs [29,14,51]. For higher-order-in-time SPDEs, such as wave equations / dispersive equations there are available results regarding invariant manifolds [132] and random attractors [165,39]. In summary, a lot of progress towards solution theory of SPDEs has been made over the last several decades.…”