Browsing by Subject "math.AP"
Now showing items 120 of 28

A convergent method for linear halfspace kinetic equations
(20170423)We give a unified proof for the wellposedness of a class of linear halfspace equations with general incoming data and construct a Galerkin method to numerically resolve this type of equations in a systematic way. Our main ... 
A Mean Field Limit for the Vlasov–Poisson System
(Archive for Rational Mechanics and Analysis, 201709)© 2017, SpringerVerlag Berlin Heidelberg. We present a probabilistic proof of the mean field limit and propagation of chaos Nparticle systems in three dimensions with positive (Coulomb) or negative (Newton) 1/r potentials ... 
A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations
This paper concerns the a priori generalization analysis of the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular neuralnetworkbased method for solving high dimensional partial differential equations. We derive ... 
A Variation on the DonskerVaradhan Inequality for the Principial Eigenvalue
(20170423)The purpose of this short note is to give a variation on the classical DonskerVaradhan inequality, which bounds the first eigenvalue of a secondorder elliptic operator on a bounded domain $\Omega$ by the largest mean first ... 
An isoperimetric problem with Coulomb repulsion and attraction to a background nucleus
(20170423)We study an isoperimetric problem the energy of which contains the perimeter of a set, Coulomb repulsion of the set with itself, and attraction of the set to a background nucleus as a point charge with charge $Z$. For the ... 
Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model
(20170423)In the absence of external material deposition, crystal surfaces usually relax to become flat by decreasing their free energy. We study an asymmetry in the relaxation of macroscopic plateaus, facets, of a periodic surface ... 
Asymptotic behavior of branching diffusion processes in periodic media
We study the asymptotic behavior of branching diffusion processes in periodic media. For a supercritical branching process, we distinguish two types of behavior for the normalized number of particles in a bounded domain, ... 
Bloch dynamics with second order Berry phase correction
(20170423)We derive the semiclassical Bloch dynamics with the second order Berry phase correction, based on a twoscale WKB asymptotic analysis. For uniform external electric field, the bicharacteristics system after a positional ... 
Convergence of PhaseField Free Energy and Boundary Force for Molecular Solvation
(20170423)We study a phasefield variational model for the solvaiton of charged molecules with an implicit solvent. The solvation freeenergy functional of all phase fields consists of the surface energy, solute excluded volume and ... 
Defect resonances of truncated crystal structures
Defects in the atomic structure of crystalline materials may spawn electronic bound states, known as \emph{defect states}, which decay rapidly away from the defect. Simplified models of defect states typically assume the ... 
Derivation of the Bogoliubov Time Evolution for Gases with Finite Speed of Sound
The derivation of meanfield limits for quantum systems at zero temperature has attracted many researchers in the last decades. Recent developments are the consideration of pair correlations in the effective description, ... 
Exterior Differential Systems and EulerLagrange Partial Differential Equations
We use methods from exterior differential systems (EDS) to develop a geometric theory of scalar, firstorder Lagrangian functionals and their associated EulerLagrange PDEs, subject to contact transformations. The first ... 
Global regularity for 1D Eulerian dynamics with singular interaction forces
(20171218)The EulerPoissonAlignment (EPA) system appears in mathematical biology and is used to model, in a hydrodynamic limit, a set agents interacting through mutual attraction/repulsion as well as alignment forces. We consider ... 
Irreducible GinzburgLandau fields in dimension 2
(20180118)GinzburgLandau fields are the solutions of the GinzburgLandau equations which depend on two positive parameters, $\alpha$ and $\beta$. We give conditions on $\alpha$ and $\beta$ for the existence of irreducible solutions ... 
Learning interacting particle systems: diffusion parameter estimation for aggregation equations
(20180214)In this article, we study the parameter estimation of interacting particle systems subject to the Newtonian aggregation. Specifically, we construct an estimator $\widehat{\nu}$ with partial observed data to approximate the ... 
Microscopic derivation of the KellerSegel equation in the subcritical regime
We derive the twodimensional KellerSegel equation from a stochastic system of $N$ interacting particles in the case of subcritical chemosensitivity $\chi < 8 \pi$. The Coulomb interaction force is regularised with a cutoff ... 
NonConvex Planar Harmonic Maps
We formulate a novel characterization of a family of invertible maps between twodimensional domains. Our work follows two classic results: The Rad\'oKneserChoquet (RKC) theorem, which establishes the invertibility of harmonic ... 
On explicit $L^2$convergence rate estimate for piecewise deterministic Markov processes
We establish $L^2$exponential convergence rate for three popular piecewise deterministic Markov processes for sampling: the randomized Hamiltonian Monte Carlo method, the zigzag process, and the bouncy particle sampler. ... 
On explicit $L^2$convergence rate estimate for underdamped Langevin dynamics
We provide a new explicit estimate of exponential decay rate of underdamped Langevin dynamics in $L^2$ distance. To achieve this, we first prove a Poincar\'{e}type inequality with Gibbs measure in space and Gaussian measure in ... 
On the meanfield limit for the VlasovPoissonFokkerPlanck system
We devise and study a random particle blob method for approximating the VlasovPoissonFokkkerPlanck (VPFP) equations by a $N$particle system subject to the Brownian motion in $\mathbb{R}^3$ space. More precisely, we show ...