seminar: Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain
speaker: Tomás Caraballo (Universidad de Sevilla)abstract:
We consider a system of semi-linear parabolic stochastic partial differential equations with
additive space-time noise on the union of thin bounded tubular domains
$D{1,\eps}$ $:=$ $\Gamma\times(0,\eps)$ and $D{2,\eps}$ $:=$ $\Gamma\times(-\eps,0)$
joined at the common base $\Gamma$ $\subset$ ${\R}{d}$ where $d\ge1$.
The equations are coupled by an interface condition on $\Gamma$ which
involves a reaction intensity $k(x',\eps)$, where $x$ $=$ $(x',x{d+1})$ $\in$
$\mathbb{R}{d+1}$ with $x'$ $\in$ $\Gamma$ and $
x{d+1}
$ $<$ $\eps$. Random
influences are included through additive space-time Brownian motion, which depend
only on the base spatial variable $x'$ $\in$ $\Gamma$ and not on the spatial
variable $x{d+1}$ in the thin direction. We will establish limiting properties of the global random
attractor as the thinness
parameter of the domain $\eps$ $\to$ $0$, i.e. as the initial domain becomes thinner,
when the intensity function possesses the property
$\lim{\eps\to0}\eps{-1}k(x',\eps)=+\infty$ In particular, the limiting dynamics is
described by a single stochastic parabolic equation with the averaged diffusion
coefficient, and nonlinearity term, which essentially indicates synchronization of the
dynamics on both sides of the common base $\Gamma$.
timetable:
Wed 29 Mar, 11:50 - 12:30, Sala Conferenze Centro De Giorgi
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