This is an expository paper on the theory of gradient flows, and in particular of those PDEs
which can be interpreted as gradient flows for the Wasserstein metric on the space of …

As a counterpoint to classical stochastic particle methods for diffusion, we develop a
deterministic particle method for linear and nonlinear diffusion. At first glance, deterministic …

Motivated by the occurrence in rate functions of time-dependent large-deviation principles,
we study a class of non-negative functions ℒ that induce a flow, given by ℒ (ρ t, ρ ̇ t)= 0 …

Abstract Stein Variational Gradient Descent (SVGD), a popular sampling algorithm, is often
described as the kernelized gradient flow for the Kullback-Leibler divergence in the …

We show that degenerate nonlinear diffusion equations can be asymptotically obtained as a
limit from a class of nonlocal partial differential equations. The nonlocal equations are …

We investigate the long time behavior of the critical mass Patlak–Keller–Segel equation.
This equation has a one parameter family of steady-state solutions ϱλ, λ> 0, with thick tails …

This chapter reviews different numerical methods for specific examples of Wasserstein
gradient flows: we focus on nonlinear Fokker-Planck equations, but also discuss …

We develop a set of numerical schemes for the Poisson–Nernst–Planck equations. We
prove that our schemes are mass conservative, uniquely solvable and keep positivity …

This paper studies the derivation of the quadratic porous medium equation and a class of
cross-diffusion systems from nonlocal interactions. We prove convergence of solutions of a …

We provide a rigorous mathematical framework to establish the limit of a nonlocal model of
cell-cell adhesion system to a local model. When the parameter of the nonlocality goes to 0 …