10,410 research outputs found
Additive noise effects in active nonlinear spatially extended systems
We examine the effects of pure additive noise on spatially extended systems
with quadratic nonlinearities. We develop a general multiscale theory for such
systems and apply it to the Kuramoto-Sivashinsky equation as a case study. We
first focus on a regime close to the instability onset (primary bifurcation),
where the system can be described by a single dominant mode. We show
analytically that the resulting noise in the equation describing the amplitude
of the dominant mode largely depends on the nature of the stochastic forcing.
For a highly degenerate noise, in the sense that it is acting on the first
stable mode only, the amplitude equation is dominated by a pure multiplicative
noise, which in turn induces the dominant mode to undergo several critical
state transitions and complex phenomena, including intermittency and
stabilisation, as the noise strength is increased. The intermittent behaviour
is characterised by a power-law probability density and the corresponding
critical exponent is calculated rigorously by making use of the first-passage
properties of the amplitude equation. On the other hand, when the noise is
acting on the whole subspace of stable modes, the multiplicative noise is
corrected by an additive-like term, with the eventual loss of any stabilised
state. We also show that the stochastic forcing has no effect on the dominant
mode dynamics when it is acting on the second stable mode. Finally, in a regime
which is relatively far from the instability onset, so that there are two
unstable modes, we observe numerically that when the noise is acting on the
first stable mode, both dominant modes show noise-induced complex phenomena
similar to the single-mode case
Defects and boundary layers in non-Euclidean plates
We investigate the behavior of non-Euclidean plates with constant negative
Gaussian curvature using the F\"oppl-von K\'arm\'an reduced theory of
elasticity. Motivated by recent experimental results, we focus on annuli with a
periodic profile. We prove rigorous upper and lower bounds for the elastic
energy that scales like the thickness squared. In particular we show that are
only two types of global minimizers -- deformations that remain flat and saddle
shaped deformations with isolated regions of stretching near the edge of the
annulus. We also show that there exist local minimizers with a periodic profile
that have additional boundary layers near their lines of inflection. These
additional boundary layers are a new phenomenon in thin elastic sheets and are
necessary to regularize jump discontinuities in the azimuthal curvature across
lines of inflection. We rigorously derive scaling laws for the width of these
boundary layers as a function of the thickness of the sheet
Stationary solutions of liquid two-layer thin film models
We investigate stationary solutions of a thin-film model for liquid two-layer
flows in an energetic formulation that is motivated by its gradient flow
structure. The goal is to achieve a rigorous understanding of the contact-angle
conditions for such two-layer systems. We pursue this by investigating a
corresponding energy that favors the upper liquid to dewet from the lower
liquid substrate, leaving behind a layer of thickness . After proving
existence of stationary solutions for the resulting system of thin-film
equations we focus on the limit via matched asymptotic analysis.
This yields a corresponding sharp-interface model and a matched asymptotic
solution that includes logarithmic switch-back terms. We compare this with
results obtained using -convergence, where we establish existence and
uniqueness of energetic minimizers in that limit
The Inviscid Limit and Boundary Layers for Navier-Stokes Flows
The validity of the vanishing viscosity limit, that is, whether solutions of
the Navier-Stokes equations modeling viscous incompressible flows converge to
solutions of the Euler equations modeling inviscid incompressible flows as
viscosity approaches zero, is one of the most fundamental issues in
mathematical fluid mechanics. The problem is classified into two categories:
the case when the physical boundary is absent, and the case when the physical
boundary is present and the effect of the boundary layer becomes significant.
The aim of this article is to review recent progress on the mathematical
analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of
Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final
publication is available at http://www.springerlink.co
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