Stochastic Analysis, Stochastic Partial Differential Equations and Applications to Fluid Dynamics and Particle Systems

# 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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