The Monge problem in Wiener Space

We address the Monge problem in the abstract Wiener space and we give an existence result provided both marginal measures are absolutely continuous with respect to the infinite dimensional Gaussian measure {\gamma}

A model for the quasi-static growth of brittle fractures based on local minimization

We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo. The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the $L^2$-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Francfort and Marigo and in our previous paper, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith's criterion holds at the crack tips. We prove also that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough.Comment: 20 page

Geodesics in the space of measure-preserving maps and plans

We study Brenier's variational models for incompressible Euler equations. These models give rise to a relaxation of the Arnold distance in the space of measure-preserving maps and, more generally, measure-preserving plans. We analyze the properties of the relaxed distance, we show a close link between the Lagrangian and the Eulerian model, and we derive necessary and sufficient optimality conditions for minimizers. These conditions take into account a modified Lagrangian induced by the pressure field. Moreover, adapting some ideas of Shnirelman, we show that, even for non-deterministic final conditions, generalized flows can be approximated in energy by flows associated to measure-preserving maps

Precise determination of muon and electromagnetic shower contents from shower universality property

We consider two new aspects of Extensive Air Shower development universality allowing to make accurate estimation of muon and electromagnetic (EM) shower contents in two independent ways. In the first case, to get muon (or EM) signal in water Cherenkov tanks or in scintillator detectors it is enough to know the vertical depth of shower maximum and the total signal in the ground detector. In the second case, the EM signal can be calculated from the primary particle energy and the zenith angle. In both cases the parametrizations of muon and EM signals are almost independent on primary particle nature, energy and zenith angle. Implications of the considered properties for mass composition and hadronic interaction studies are briefly discussed. The present study is performed on 28000 of proton, oxygen and iron showers, generated with CORSIKA 6.735 for $E^{-1}$ spectrum in the energy range log(E/eV)=18.5-20.0 and uniformly distributed in cos^2(theta) in zenith angle interval theta=0-65 degrees for QGSJET II/Fluka interaction models.Comment: Submitted to Phys. Rev.

Fokker-Planck type equations with Sobolev diffusion coefficients and BV drift coefficients

In this paper we give an affirmative answer to an open question mentioned in [Le Bris and Lions, Comm. Partial Differential Equations 33 (2008), 1272--1317], that is, we prove the well-posedness of the Fokker-Planck type equations with Sobolev diffusion coefficients and BV drift coefficients.Comment: 11 pages. The proof has been modifie

Perimeter of sublevel sets in infinite dimensional spaces

We compare the perimeter measure with the Airault-Malliavin surface measure and we prove that all open convex subsets of abstract Wiener spaces have finite perimeter. By an explicit counter-example, we show that in general this is not true for compact convex domains

Large Deviations in Stochastic Heat-Conduction Processes Provide a Gradient-Flow Structure for Heat Conduction

We consider three one-dimensional continuous-time Markov processes on a lattice, each of which models the conduction of heat: the family of Brownian Energy Processes with parameter $m$, a Generalized Brownian Energy Process, and the Kipnis-Marchioro-Presutti process. The hydrodynamic limit of each of these three processes is a parabolic equation, the linear heat equation in the case of the BEP$(m)$ and the KMP, and a nonlinear heat equation for the GBEP($a$). We prove the hydrodynamic limit rigorously for the BEP$(m)$, and give a formal derivation for the GBEP($a$). We then formally derive the pathwise large-deviation rate functional for the empirical measure of the three processes. These rate functionals imply gradient-flow structures for the limiting linear and nonlinear heat equations. We contrast these gradient-flow structures with those for processes describing the diffusion of mass, most importantly the class of Wasserstein gradient-flow systems. The linear and nonlinear heat-equation gradient-flow structures are each driven by entropy terms of the form $-\log \rho$; they involve dissipation or mobility terms of order $\rho^2$ for the linear heat equation, and a nonlinear function of $\rho$ for the nonlinear heat equation.Comment: 29 page

Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case

In this paper we use the new regularity and stability estimates for Alexandrov solutions to Monge-Ampere equations estabilished by G.De Philippis and A.Figalli to provide a global in time existence of distributional solutions to a semigeostrophic equation on the 2-dimensional torus, under very mild assumptions on the initial data. A link with Lagrangian solutions is also discussed.Comment: 16 pages, no figure

A JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows

In this article we set up a splitting variant of the JKO scheme in order to handle gradient flows with respect to the Kantorovich-Fisher-Rao metric, recently introduced and defined on the space of positive Radon measure with varying masses. We perform successively a time step for the quadratic Wasserstein/Monge-Kantorovich distance, and then for the Hellinger/Fisher-Rao distance. Exploiting some inf-convolution structure of the metric we show convergence of the whole process for the standard class of energy functionals under suitable compactness assumptions, and investigate in details the case of internal energies. The interest is double: On the one hand we prove existence of weak solutions for a certain class of reaction-advection-diffusion equations, and on the other hand this process is constructive and well adapted to available numerical solvers.Comment: Final version, to appear in SIAM SIM