Random Attractor for Stochastic Wave Equation with Arbitrary Exponent and Additive Noise on Rn\mathbb{R}^n

Asymptotic random dynamics of weak solutions for a damped stochastic wave equation with the nonlinearity of arbitrarily large exponent and the additive noise on $\mathbb{R}^n$ is investigated. The existence of a pullback random attractor is proved in a parameter region with a breakthrough in proving the pullback asymptotic compactness of the cocycle with the quasi-trajectories defined on the integrable function space of arbitrary exponent and on the unbounded domain of arbitrary dimension

Випадковий атрактор напівлінійного стохастичного збуреного хвильового рівняння без одиничності розв’язку

Досліджено динаміку розв’язків напівлінійного хвильового рівняння, збуреного адитивним білим шумом, із точки зору теорії випадкових атракторів. Умови на параметри задачі не гарантують єдності розв’язку відповідної задачі Коші. Доведено теорему про існування випадкового атрактора для абстрактної некомпактної багатозначної випадкової динамічної системи, що була застосована до хвильового рівняння з негладким нелінійним доданком. Встановлено апріорну оцінку для слабкого розв’язку випадково збуреної задачі, що дозволило отримати існування принаймні одного слабкого розв’язку. На слабких розв’язках досліджуваної задачі побудовано багатозначний стохастичний потік. Доведено існування випадкового атрактора для побудованого багатозначного стохастичного потоку.In this paper we investigate the dynamics of solutions of the semilinear wave equation, perturbed by additive white noise, in sense of the random attractor theory. The conditions on the parameters of the problem do not guarantee uniqueness of solution of the corresponding Cauchy problem. We prove theorem on the existence of random attractor for abstract noncompact multi-valued random dynamical system, which is applied to the wave equation with non-smooth nonlinear term. A priory estimate for weak solution of randomly perturbed problem is deduced, which allows to obtain the existence at least one weak solution. The multi-valued stochastic flow is generated by the weak solutions of investigated problem. We prove the existence of random attractor for generated multi-valued stochastic flow.Исследована динамика решений полулинейного волнового уравнения, возмущенного аддитивным белым шумом, с точки зрения теории случайных аттракторов. Условия на параметры задачи не гарантируют единственности решения соответствующей задачи Коши. Доказано теорему о существовании случайного аттрактора для абстрактной некомпактной многозначной случайной динамической системы, которая была применена к волновому уравнению с негладким нелинейным слагаемым. Установлена априорная оценка для слабого решения случайно возмущенной задачи, которая позволила получить существование, по крайней мере, одного слабого решения. На слабых решениях исследованной задачи построен многозначный стохастический поток. Доказано существование случайного аттрактора для построенного многозначного стохастического потока

Rigorous results in space-periodic two-dimensional turbulence

We survey the recent advance in the rigorous qualitative theory of the 2d stochastic Navier-Stokes system that are relevant to the description of turbulence in two-dimensional fluids. After discussing briefly the initial-boundary value problem and the associated Markov process, we formulate results on the existence, uniqueness and mixing of a stationary measure. We next turn to various consequences of these properties: strong law of large numbers, central limit theorem, and random attractors related to a unique stationary measure. We also discuss the Donsker-Varadhan and Freidlin-Wentzell type large deviations, as well as the inviscid limit and asymptotic results in 3d thin domains. We conclude with some open problems

Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domains

This paper is concerned with the asymptotic behavior of solutions to a class of non-autonomous stochastic nonlinear wave equations with dispersive and viscosity dissipative terms driven by operator-type noise defined on the entire space Rn. The existence, uniqueness, time-semi-uniform compactness and asymptotically autonomous robustness of pullback random attractors are proved in H1(Rn) _ H1(Rn) when the growth rate of the nonlinearity has a subcritical range, the density of the noise is suitably controllable, and the time-dependent force converges to a time-independent function in some sense. The main difficulty to establish the time-semi-uniform pullback asymptotic compactness of the solutions in H1(Rn) _ H1(Rn) is caused by the lack of compact Sobolev embeddings on Rn, as well as the weak dissipativeness of the equations is surmounted at light of the idea of uniform tail-estimates and a spectral decomposition approach. The measurability of random attractors is proved by using an argument which considers two attracting universes developed by Wang and Li (Phys. D 382: 46-57, 2018)

Cooperative surmounting of bottlenecks

The physics of activated escape of objects out of a metastable state plays a key role in diverse scientific areas involving chemical kinetics, diffusion and dislocation motion in solids, nucleation, electrical transport, motion of flux lines superconductors, charge density waves, and transport processes of macromolecules, to name but a few. The underlying activated processes present the multidimensional extension of the Kramers problem of a single Brownian particle. In comparison to the latter case, however, the dynamics ensuing from the interactions of many coupled units can lead to intriguing novel phenomena that are not present when only a single degree of freedom is involved. In this review we report on a variety of such phenomena that are exhibited by systems consisting of chains of interacting units in the presence of potential barriers. In the first part we consider recent developments in the case of a deterministic dynamics driving cooperative escape processes of coupled nonlinear units out of metastable states. The ability of chains of coupled units to undergo spontaneous conformational transitions can lead to a self-organised escape. The mechanism at work is that the energies of the units become re-arranged, while keeping the total energy conserved, in forming localised energy modes that in turn trigger the cooperative escape. We present scenarios of significantly enhanced noise-free escape rates if compared to the noise-assisted case. The second part deals with the collective directed transport of systems of interacting particles overcoming energetic barriers in periodic potential landscapes. Escape processes in both time-homogeneous and time-dependent driven systems are considered for the emergence of directed motion. It is shown that ballistic channels immersed in the associated high-dimensional phase space are the source for the directed long-range transport