arXiv subjects

M. G. Delgadino

Publications and source records attributed to M. G. Delgadino.

9 recordsLinked to original sources

arXiv2025

Contractivity of Wasserstein distance and exponential decay for the Landau equation with Maxwellian molecules

Following the breakthrough work of Guillen and Silvestre \cite{GS24}, that shows that the Fisher information is monotonically decreasing for solutions to the homogeneous Landau equation, we study, for the same equation, the monotonicity properties of other physically relevant functionals. In the case of Maxwellian molecules, we show that the relative $L^2$ norm with respect to the equilibrium decays exponentially fast in time and is monotonically decreasing after some time. Moreover, still for the Maxwellian case, we provide a novel and short quantitative proof of time monotonicity of the entropic Wasserstein metric. For soft potentials, we show that the Wasserstein metric is contractive, conditional to $L^1(0,T,L^p(\mathbb{R}^3))$ bound for the solution. This result provides an alternative proof of the Fournier and Fournier-Guerin uniqueness theorem in \cite{fournier2009well_posedness_soft_potentials} \cite{fournier2010uniqueness_Coulomb}

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arXiv2024

GAN: Dynamics

We study quantitatively the overparametrization limit of the original Wasserstein-GAN algorithm. Effectively, we show that the algorithm is a stochastic discretization of a system of continuity equations for the parameter distributions of the generator and discriminator. We show that parameter clipping to satisfy the Lipschitz condition in the algorithm induces a discontinuous vector field in the mean field dynamics, which gives rise to blow-up in finite time of the mean field dynamics. We look into a specific toy example that shows that all solutions to the mean field equations converge in the long time limit to time periodic solutions, this helps explain the failure to converge.

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arXiv2021

Fast Diffusion leads to partial mass concentration in Keller-Segel type stationary solutions

We show that partial mass concentration can happen for stationary solutions of aggregation-diffusion equations with homogeneous attractive kernels in the fast diffusion range. More precisely, we prove that the free energy admits a radial global minimizer in the set of probability measures which may have part of its mass concentrated in a Dirac delta at a given point. In the case of the quartic interaction potential, we find the exact range of the diffusion exponent where concentration occurs in space dimensions $N\geq6$. We then provide numerical computations which suggest the occurrence of mass concentration in all dimensions $N\geq3$, for homogeneous interaction potentials with higher power.

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arXiv2019

A proof of the mean-field limit for $λ$-convex potentials by $Γ$-Convergence

In this work we give a proof of the mean-field limit for $λ$-convex potentials using a purely variational viewpoint. Our approach is based on the observation that all evolution equations that we study can be written as gradient flows of functionals at different levels: in the set of probability measures, in the set of symmetric probability measures on $N$ variables, and in the set of probability measures on probability measures. This basic fact allows us to rely on $Γ$-convergence tools for gradient flows to complete the proof by identifying the limits of the different terms in the Evolutionary Variational Inequalities (EVIs) associated to each gradient flow. The $λ$-convexity of the confining and interaction potentials is crucial for the unique identification of the limits and for deriving the EVIs at each description level of the interacting particle system.

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arXiv2018

Free Energies and the Reversed HLS Inequality

We prove reversed Hardy-Littlewood-Sobolev inequalities by carefully studying the natural associated free energies with direct methods of calculus of variations. Tightness is obtained by a dyadic argument, which quantifies the relative strength of the entropy functional versus the interaction energy. The existence of optimizers is shown in the class of $\prob$. With respect to their regularity, we study conditions for optimizers to be bounded functions. In a related model, we show the condensation phenomena, which suggests that optimizers are not in general regular.

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arXiv2018

Existence of ground states for aggregation-diffusion equations

We analyze free energy functionals for macroscopic models of multi-agent systems interacting via pairwise attractive forces and localized repulsion. The repulsion at the level of the continuous description is modeled by pressure-related terms in the functional making it energetically favorable to spread, while the attraction is modeled through nonlocal forces. We give conditions on general entropies and interaction potentials for which neither ground states nor local minimizers exist. We show that these results are sharp for homogeneous functionals with entropies leading to degenerate diffusions while they are not sharp for fast diffusions. The particular relevant case of linear diffusion is totally clarified giving a sharp condition on the interaction potential under which the corresponding free energy functional has ground states or not.

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arXiv2017

Alexandrov's theorem revisited

We show that among sets of finite perimeter balls are the only volume-constrained critical points of the perimeter functional.

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arXiv2017

Bubbling with $L^2$-almost constant mean curvature and an Alexandrov-type theorem for crystals

A compactness theorem for volume-constrained almost-critical points of elliptic integrands is proven. The result is new even for the area functional, as almost-criticality is measured in an integral rather than in a uniform sense. Two main applications of the compactness theorem are discussed. First, we obtain a description of critical points/local minimizers of elliptic energies interacting with a confinement potential. Second, we prove an Alexandrov-type theorem for crystalline isoperimetric problems.

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arXiv2014

Regularity of local minimizers of the interaction energy via obstacle problems

The repulsion strength at the origin for repulsive/attractive potentials determines the regularity of local minimizers of the interaction energy. In this paper, we show that if this repulsion is like Newtonian or more singular than Newtonian (but still locally integrable), then the local minimizers must be locally bounded densities (and even continuous for more singular than Newtonian repulsion). We prove this (and some other regularity results) by first showing that the potential function associated to a local minimizer solves an obstacle problem and then by using classical regularity results for such problems.

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