62 research outputs found
On the rate of convergence in periodic homogenization of scalar first-order ordinary differential equations
In this paper, we study the rate of convergence in periodic homogenization of
scalar ordinary differential equations. We provide a quantitative error
estimate between the solutions of a first-order ordinary differential equation
with rapidly oscillating coefficients and the limiting homogenized solution. As
an application of our result, we obtain an error estimate for the solution of
some particular linear transport equations
Aubry sets for weakly coupled systems of Hamilton--Jacobi equations
We introduce a notion of Aubry set for weakly coupled systems of
Hamilton--Jacobi equations on the torus and characterize it as the region where
the obstruction to the existence of globally strict critical subsolutions
concentrates. As in the case of a single equation, we prove the existence of
critical subsolutions which are strict and smooth outside the Aubry set. This
allows us to derive in a simple way a comparison result among critical sub and
supersolutions with respect to their boundary data on the Aubry set, showing in
particular that the latter is a uniqueness set for the critical system. We also
highlight some rigidity phenomena taking place on the Aubry set.Comment: 35 pages v.2 the introduction has been rewritten and shortened. Some
proofs simplified. Corrections and references added. Corollary 5.3 added
stating antisymmetry of the Ma\~n\'e matrix on points of the Aubry set.
Section 6 contains a new example
Homogenization of linear transport equations in a stationary ergodic setting
We study the homogenization of a linear kinetic equation which models the
evolution of the density of charged particles submitted to a highly oscillating
electric field. The electric field and the initial density are assumed to be
random and stationary. We identify the asymptotic microscopic and macroscopic
profiles of the density, and we derive formulas for these profiles when the
space dimension is equal to one.Comment: 24 page
Homogenization and enhancement for the G-equation
We consider the so-called G-equation, a level set Hamilton-Jacobi equation,
used as a sharp interface model for flame propagation, perturbed by an
oscillatory advection in a spatio-temporal periodic environment. Assuming that
the advection has suitably small spatial divergence, we prove that, as the size
of the oscillations diminishes, the solutions homogenize (average out) and
converge to the solution of an effective anisotropic first-order
(spatio-temporal homogeneous) level set equation. Moreover we obtain a rate of
convergence and show that, under certain conditions, the averaging enhances the
velocity of the underlying front. We also prove that, at scale one, the level
sets of the solutions of the oscillatory problem converge, at long times, to
the Wulff shape associated with the effective Hamiltonian. Finally we also
consider advection depending on position at the integral scale
A Sublinear Variance Bound for Solutions of a Random Hamilton Jacobi Equation
We estimate the variance of the value function for a random optimal control
problem. The value function is the solution of a Hamilton-Jacobi
equation with random Hamiltonian
in dimension . It is known that homogenization occurs as , but little is known about the statistical fluctuations of .
Our main result shows that the variance of the solution is bounded
by . The proof relies on a modified Poincar\'e
inequality of Talagrand
Proceedings of minisemester on evolution of interfaces, Sapporo 2010
conf: Special Project A, Proceedings of minisemester on evolution of interfaces, Sapporo (Department of Mathematics, Hokkaido University, July 12- August 13, 2010
The one-dimensional Keller-Segel model with fractional diffusion of cells
We investigate the one-dimensional Keller-Segel model where the diffusion is
replaced by a non-local operator, namely the fractional diffusion with exponent
. We prove some features related to the classical
two-dimensional Keller-Segel system: blow-up may or may not occur depending on
the initial data. More precisely a singularity appears in finite time when
and the initial configuration of cells is sufficiently concentrated.
On the opposite, global existence holds true for if the initial
density is small enough in the sense of the norm.Comment: 12 page
Repeated games for eikonal equations, integral curvature flows and non-linear parabolic integro-differential equations
The main purpose of this paper is to approximate several non-local evolution
equations by zero-sum repeated games in the spirit of the previous works of
Kohn and the second author (2006 and 2009): general fully non-linear parabolic
integro-differential equations on the one hand, and the integral curvature flow
of an interface (Imbert, 2008) on the other hand. In order to do so, we start
by constructing such a game for eikonal equations whose speed has a
non-constant sign. This provides a (discrete) deterministic control
interpretation of these evolution equations. In all our games, two players
choose positions successively, and their final payoff is determined by their
positions and additional parameters of choice. Because of the non-locality of
the problems approximated, by contrast with local problems, their choices have
to "collect" information far from their current position. For integral
curvature flows, players choose hypersurfaces in the whole space and positions
on these hypersurfaces. For parabolic integro-differential equations, players
choose smooth functions on the whole space
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