141 research outputs found
Modulational stability of cellular flows
We present here the homogenization of the equations for the initial modulational (large-scale) perturbations of stationary solutions of the two-dimensional NavierâStokes equations with a time-independent periodic rapidly oscillating forcing. The stationary solutions are cellular flows and they are determined by the stream function phi = sinx1/epsilonsinx2/epsilon+δ cosx1/epsiloncosx2/epsilon, 0 ⤠δ ⤠1. Two results are given here. For any Reynolds number we prove the homogenization of the linearized equations. For small Reynolds number we prove the homogenization for the fully nonlinear problem. These results show that the modulational stability of cellular flows is determined by the stability of the effective (homogenized) equations
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
Rise of correlations of transformation strains in random polycrystals
We investigate the statistics of the transformation strains that arise in random martensitic polycrystals as boundary conditions cause its component crystallites to undergo martensitic phase transitions. In our laminated polycrystal model the orientation of the n grains (crystallites)
is given by an uncorrelated random array of the orientation angles θ_i, i = 1, . . . ,n. Under imposed boundary conditions the polycrystal grains may undergo a martensitic transformation. The associated transformation strains ξ_i, i = 1, . . . ,n depend on the array of orientation angles, and they can be obtained as a solution to a nonlinear optimization problem. While the random variables θ_i,
i = 1, . . . ,n are uncorrelated, the random variables Îľ_i, i = 1, . . . ,n may be correlated. This issue is
central in our considerations. We investigate it in following three different scaling limits: (i) Infinitely
long grains (laminated polycrystal of height L = â); (ii) Grains of finite but large height (L Âť 1); and (iii) Chain of short grains (L = l_0/(2n), l_0 ÂŤ 1). With references to de Finettiâs theorem, Rieszâ
rearrangement inequality, and near neighbor approximations, our analyses establish that under the
scaling limits (i), (ii), and (iii) the arrays of transformation strains arising from given boundary
conditions exhibit no correlations, long-range correlations, and exponentially decaying short-range
correlations, respectivel
Illumination strategies for intensity-only imaging
We propose a new strategy for narrow band, active array imaging of localized
scat- terers when only the intensities are recorded and measured at the array.
We consider a homogeneous medium so that wave propagation is fully coherent. We
show that imaging with intensity-only measurements can be carried out using the
time reversal operator of the imaging system, which can be obtained from
intensity measurements using an appropriate illumination strategy and the
polarization identity. Once the time reversal operator has been obtained, we
show that the images can be formed using its singular value decomposition
(SVD). We use two SVD-based methods to image the scatterers. The proposed
approach is simple and efficient. It does not need prior information about the
sought image, and guarantees exact recovery in the noise-free case.
Furthermore, it is robust with respect to additive noise. Detailed numerical
simulations illustrate the performance of the proposed imaging strategy when
only the intensities are captured
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