93 research outputs found
Mean-field limits for some Riesz interaction gradient flows
This paper is concerned with the mean-field limit for the gradient flow
evolution of particle systems with pairwise Riesz interactions, as the number
of particles tends to infinity. Based on a modulated energy method, using
regularity and stability properties of the limiting equation, as inspired by
the work of Serfaty in the context of the Ginzburg-Landau vortices, we prove a
mean-field limit result in dimensions 1 and 2 in cases for which this problem
was still open
Well-posedness for mean-field evolutions arising in superconductivity
We establish the existence of a global solution for a new family of
fluid-like equations, which are obtained in a joint work with Serfaty in
certain regimes as the mean-field evolution of the supercurrent density in a
(2D section of a) type-II superconductor with pinning and with imposed electric
current. We also consider general vortex-sheet initial data, and investigate
the uniqueness and regularity properties of the solution. For some choice of
parameters, the equation under investigation coincides with the so-called lake
equation from 2D shallow water fluid dynamics, and our analysis then leads to a
new existence result for rough initial data.Comment: 53 pages; revised version, including an appendix jointly written with
Julian Fischer about global existence for the degenerate parabolic cas
Higher-order pathwise theory of fluctuations in stochastic homogenization
We consider linear elliptic equations in divergence form with stationary
random coefficients of integrable correlations. We characterize the
fluctuations of a macroscopic observable of a solution to relative order
, where is the spatial dimension; the fluctuations turn out to
be Gaussian. As for previous work on the leading order, this higher-order
characterization relies on a pathwise proximity of the macroscopic fluctuations
of a general solution to those of the (higher-order) correctors, via a
(higher-order) two-scale expansion injected into the homogenization commutator,
thus confirming the scope of this notion. This higher-order generalization
sheds a clearer light on the algebraic structure of the higher-order versions
of correctors, flux correctors, two-scale expansions, and homogenization
commutators. It reveals that in the same way as this algebra provides a
higher-order theory for microscopic spatial oscillations, it also provides a
higher-order theory for macroscopic random fluctuations, although both
phenomena are not directly related. We focus on the model framework of an
underlying Gaussian ensemble, which allows for an efficient use of
(second-order) Malliavin calculus for stochastic estimates. On the technical
side, we introduce annealed Calder\'on-Zygmund estimates for the elliptic
operator with random coefficients, which conveniently upgrade the known
quenched large-scale estimates.Comment: 57 page
Stochastic homogenization of nonconvex unbounded integral functionals with convex growth
We consider the well-travelled problem of homogenization of random integral
functionals. When the integrand has standard growth conditions, the qualitative
theory is well-understood. When it comes to unbounded functionals, that is,
when the domain of the integrand is not the whole space and may depend on the
space-variable, there is no satisfactory theory. In this contribution we
develop a complete qualitative stochastic homogenization theory for nonconvex
unbounded functionals with convex growth. We first prove that if the integrand
is convex and has -growth from below (with , the dimension), then it
admits homogenization regardless of growth conditions from above. This result,
that crucially relies on the existence and sublinearity at infinity of
correctors, is also new in the periodic case. In the case of nonconvex
integrands, we prove that a similar homogenization result holds provided the
nonconvex integrand admits a two-sided estimate by a convex integrand (the
domain of which may depend on the space-variable) that itself admits
homogenization. This result is of interest to the rigorous derivation of rubber
elasticity from polymer physics, which involves the stochastic homogenization
of such unbounded functionals.Comment: 64 pages, 2 figure
Semi-dilute rheology of particle suspensions: derivation of Doi-type models
This work is devoted to the large-scale rheology of suspensions of
non-Brownian inertialess rigid particles, possibly self-propelling, suspended
in Stokes flow. Starting from a hydrodynamic model, we derive a semi-dilute
mean-field description in form of a Doi-type model, which is given by a
'macroscopic' effective Stokes equation coupled with a 'microscopic' Vlasov
equation for the statistical distribution of particle positions and
orientations. This accounts for some non-Newtonian effects since the viscosity
in the effective Stokes equation depends on the local distribution of particle
orientations via Einstein's formula. The main difficulty is the detailed
analysis of multibody hydrodynamic interactions between the particles, which we
perform by means of a cluster expansion combined with a multipole expansion in
a suitable dilute regime.Comment: 47 page
Effective viscosity of random suspensions without uniform separation
This work is devoted to the definition and the analysis of the effective
viscosity associated with a random suspension of small rigid particles in a
steady Stokes fluid. While previous works on the topic have been conveniently
assuming that particles are uniformly separated, we relax this restrictive
assumption in form of mild moment bounds on interparticle distances.Comment: 37 page
Analyticity of homogenized coefficients under Bernoulli perturbations and the Clausius-Mossotti formulas
This paper is concerned with the behavior of the homogenized coefficients
associated with some random stationary ergodic medium under a Bernoulli
perturbation. Introducing a new family of energy estimates that combine
probability and physical spaces, we prove the analyticity of the perturbed
homogenized coefficients with respect to the Bernoulli parameter. Our approach
holds under the minimal assumptions of stationarity and ergodicity, both in the
scalar and vector cases, and gives analytical formulas for each derivative that
essentially coincide with the so-called cluster expansion used by physicists.
In particular, the first term yields the celebrated (electric and elastic)
Clausius-Mossotti formulas for isotropic spherical random inclusions in an
isotropic reference medium. This work constitutes the first general proof of
these formulas in the case of random inclusions.Comment: 47 page
Multiscale functional inequalities in probability: Concentration properties
In a companion article we have introduced a notion of multiscale functional
inequalities for functions of an ergodic stationary random field on
the ambient space . These inequalities are multiscale weighted
versions of standard Poincar\'e, covariance, and logarithmic Sobolev
inequalities. They hold for all the examples of fields arising in the
modelling of heterogeneous materials in the applied sciences whereas their
standard versions are much more restrictive. In this contribution we first
investigate the link between multiscale functional inequalities and more
standard decorrelation or mixing properties of random fields. Next, we show
that multiscale functional inequalities imply fine concentration properties for
nonlinear functions . This constitutes the main stochastic ingredient to
the quenched large-scale regularity theory for random elliptic operators by the
second author, Neukamm, and Otto, and to the corresponding quantitative
stochastic homogenization results.Comment: 24 page
Resource dependent branching processes and the envelope of societies
Since its early beginnings, mankind has put to test many different society
forms, and this fact raises a complex of interesting questions. The objective
of this paper is to present a general population model which takes essential
features of any society into account and which gives interesting answers on the
basis of only two natural hypotheses. One is that societies want to survive,
the second, that individuals in a society would, in general, like to increase
their standard of living. We start by presenting a mathematical model, which
may be seen as a particular type of a controlled branching process. All
conditions of the model are justified and interpreted. After several
preliminary results about societies in general we can show that two society
forms should attract particular attention, both from a qualitative and a
quantitative point of view. These are the so-called weakest-first society and
the strongest-first society. In particular we prove then that these two
societies stand out since they form an envelope of all possible societies in a
sense we will make precise. This result (the envelopment theorem) is seen as
significant because it is paralleled with precise survival criteria for the
enveloping societies. Moreover, given that one of the "limiting" societies can
be seen as an extreme form of communism, and the other one as being close to an
extreme version of capitalism, we conclude that, remarkably, humanity is close
to having already tested the limits.Comment: Published in at http://dx.doi.org/10.1214/13-AAP998 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Mean Field Limit for Coulomb-Type Flows
We establish the mean-field convergence for systems of points evolving along
the gradient flow of their interaction energy when the interaction is the
Coulomb potential or a super-coulombic Riesz potential, for the first time in
arbitrary dimension. The proof is based on a modulated energy method using a
Coulomb or Riesz distance, assumes that the solutions of the limiting equation
are regular enough and exploits a weak-strong stability property for them. The
method can handle the addition of a regular interaction kernel, and applies
also to conservative and mixed flows. In the appendix, it is also adapted to
prove the mean-field convergence of the solutions to Newton's law with Coulomb
or Riesz interaction in the monokinetic case to solutions of an Euler-Poisson
type system.Comment: Final version with expanded introduction, to appear in Duke Math
Journal. 35 page
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