151 research outputs found
Anomalous diffusion in fast cellular flows at intermediate time scales
It is well known that on long time scales the behaviour of tracer particles
diffusing in a cellular flow is effectively that of a Brownian motion. This
paper studies the behaviour on "intermediate" time scales before diffusion sets
in. Various heuristics suggest that an anomalous diffusive behaviour should be
observed. We prove that the variance on intermediate time scales grows like
. Hence, on these time scales the effective behaviour can not be
purely diffusive, and is consistent with an anomalous diffusive behaviour.Comment: 28 pages, 2 figure
The regularizing effects of resetting in a particle system for the Burgers equation
We study the dissipation mechanism of a stochastic particle system for the
Burgers equation. The velocity field of the viscous Burgers and Navier-Stokes
equations can be expressed as an expected value of a stochastic process based
on noisy particle trajectories [Constantin and Iyer Comm. Pure Appl. Math. 3
(2008) 330-345]. In this paper we study a particle system for the viscous
Burgers equations using a Monte-Carlo version of the above; we consider N
copies of the above stochastic flow, each driven by independent Wiener
processes, and replace the expected value with times the sum over
these copies. A similar construction for the Navier-Stokes equations was
studied by Mattingly and the first author of this paper [Iyer and Mattingly
Nonlinearity 21 (2008) 2537-2553]. Surprisingly, for any finite N, the particle
system for the Burgers equations shocks almost surely in finite time. In
contrast to the full expected value, the empirical mean
does not regularize the system enough to ensure a time global solution. To
avoid these shocks, we consider a resetting procedure, which at first sight
should have no regularizing effect at all. However, we prove that this
procedure prevents the formation of shocks for any , and consequently
as we get convergence to the solution of the viscous Burgers
equation on long time intervals.Comment: Published in at http://dx.doi.org/10.1214/10-AOP586 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Quantifying the dissipation enhancement of cellular flows
We study the dissipation enhancement by cellular flows. Previous work by
Iyer, Xu, and Zlato\v{s} produces a family of cellular flows that can enhance
dissipation by an arbitrarily large amount. We improve this result by providing
quantitative bounds on the dissipation enhancement in terms of the flow
amplitude, cell size and diffusivity. Explicitly we show that the mixing time
is bounded by the exit time from one cell when the flow amplitude is large
enough, and by the reciprocal of the effective diffusivity when the flow
amplitude is small. This agrees with the optimal heuristics. We also prove a
general result relating the dissipation time of incompressible flows to the
mixing time. The main idea behind the proof is to study the dynamics
probabilistically and construct a successful coupling.Comment: 21 pages, 2 figure
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