64 research outputs found
Modulation Equations: Stochastic Bifurcation in Large Domains
We consider the stochastic Swift-Hohenberg equation on a large domain near
its change of stability. We show that, under the appropriate scaling, its
solutions can be approximated by a periodic wave, which is modulated by the
solutions to a stochastic Ginzburg-Landau equation. We then proceed to show
that this approximation also extends to the invariant measures of these
equations
Enstrophy Dynamics of Stochastically Forced Large-Scale Geophysical Flows
Enstrophy is an averaged measure of fluid vorticity. This quantity is
particularly important in {\em rotating} geophysical flows. We investigate the
dynamical evolution of enstrophy for large-scale quasi-geostrophic flows under
random wind forcing. We obtain upper bounds on the enstrophy, as well as
results establishing its H\"older continuity and describing the small-time
asymptotics
Multiscale Analysis for SPDEs with Quadratic Nonlinearities
In this article we derive rigorously amplitude equations for stochastic PDEs
with quadratic nonlinearities, under the assumption that the noise acts only on
the stable modes and for an appropriate scaling between the distance from
bifurcation and the strength of the noise. We show that, due to the presence of
two distinct timescales in our system, the noise (which acts only on the fast
modes) gets transmitted to the slow modes and, as a result, the amplitude
equation contains both additive and multiplicative noise.
As an application we study the case of the one dimensional Burgers equation
forced by additive noise in the orthogonal subspace to its dominant modes. The
theory developed in the present article thus allows to explain theoretically
some recent numerical observations from [Rob03]
Motion of a droplet for the mass-conserving stochastic Allen-Cahn equation
We study the stochastic mass-conserving Allen-Cahn equation posed on a
bounded two-dimensional domain with additive spatially smooth space-time noise.
This equation associated with a small positive parameter describes the
stochastic motion of a small almost semicircular droplet attached to domain's
boundary and moving towards a point of locally maximum curvature. We apply
It\^o calculus to derive the stochastic dynamics of the droplet by utilizing
the approximately invariant manifold introduced by Alikakos, Chen and Fusco for
the deterministic problem. In the stochastic case depending on the scaling, the
motion is driven by the change in the curvature of the boundary and the
stochastic forcing. Moreover, under the assumption of a sufficiently small
noise strength, we establish stochastic stability of a neighborhood of the
manifold of droplets in and , which means that with overwhelming
probability the solution stays close to the manifold for very long time-scales
Характерні особливості розвитку вітчизняних хімічних наукових шкіл
Проведено аналіз діяльності відомих учених-хіміків, які стали лідерами формування наукових шкіл у хімічних науках. Установлено актуальність їх діяльності та визначено характерні особливості розвитку вітчизняних наукових шкіл, котрі мали широкі та важливі напрямки досліджень.Activity analysis of the famous chemist-scientists, which have become leaders in the forming of scientific schools in chemistry sciences is transacted. Urgency of their activity is inserted and characters domestic scientific schools development, which had wide and important directions of the investigations are rated
A dynamical approximation for stochastic partial differential equations
Random invariant manifolds often provide geometric structures for
understanding stochastic dynamics. In this paper, a dynamical approximation
estimate is derived for a class of stochastic partial differential equations,
by showing that the random invariant manifold is almost surely asymptotically
complete. The asymptotic dynamical behavior is thus described by a stochastic
ordinary differential system on the random invariant manifold, under suitable
conditions. As an application, stationary states (invariant measures) is
considered for one example of stochastic partial differential equations.Comment: 28 pages, no figure
Markovianity and ergodicity for a surface growth PDE
The paper analyzes a model in surface growth where the uniqueness of weak solutions seems to be out of reach. We prove existence of a weak martingale solution satisfying energy inequalities and having the Markov property. Furthermore, under nondegeneracy conditions on the noise, we establish that any such solution is strong Feller and has a unique invariant measure
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