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We construct deterministic particle solutions for linear and fast diffusion equations using a nonlocal approximation. We exploit the $2$-Wasserstein gradient flow structure of the equations in order to obtain the nonlocal approximating PDEs by regularising the corresponding internal energy with suitably chosen mollifying kernels, either compactly or globally supported. Weak solutions are obtained by the JKO scheme. From the technical point of view, we improve known commutator estimates, fundamental in the nonlocal-to-local limit, to include globally supported kernels which, in particular cases, allow us to justify the limit without any further perturbation needed. Furthermore, we prove geodesic convexity of the nonlocal energies in order to prove convergence of the particle solutions to the nonlocal equations towards weak solutions of the local equations. We overcome the crucial difficulty of dealing with the singularity of the first variation of the free energies at the origin. As a byproduct, we provide convergence rates expressed as a scaling relationship between the number of particles and the localisation parameter. The analysis we perform leverages the fact that globally supported kernels yield a better convergence rate compared to compactly supported kernels. Our result is relevant in statistics, more precisely in sampling Gibbs and heavy-tailed distributions.

This paper addresses congested transport, which can be described, at macroscopic scales, by a continuity equation with a pressure variable generated from the hard-congestion constraint (maximum value of the density). The main goal of the paper is to show that, in one spatial dimension, this continuum PDE can be derived as the mean-field limit of a system of ordinary differential equations that describes the motion of a large number of particles constrained to stay at some finite distance from each others. To show that these two models describe the same dynamics at different scale, we will rely on both the Eulerian and Lagrangian points of view and use two different approximations for the density and pressure variables in the continuum limit., Comment: 23 pages, 1 figure

This paper reviews (and expands) some recent results on the modeling of aggregation-diffusion phenomena at various scales, focusing on the emergence of collective dynamics as a result of the competition between attractive and repulsive phenomena - especially (but not exclusively) in the context of attractive chemotaxis phenomena. At microscopic scales, particles (or other agents) are represented by spheres of radius $\delta>0$ and we discuss both soft-sphere models (with a pressure term penalizing the overlap of the particles) and hard-sphere models (in which overlap is prohibited). The first case leads to so-called ``blob models" which have received some attention recently as a tool to approximate non-linear diffusion by particle systems. The hard-sphere model is similar to a classical model for congested crowd motion. We review well-posedness results for these models and discuss their relationship to classical continuum description of aggregation-diffusion phenomena in the limit $\delta\to0$: the classical nonlinear drift diffusion equation and its incompressible counterpart. In the second part of the paper, we discuss recent results on the emergence and evolution of sharp interfaces when a large population of particles is considered at appropriate space and time scales: At some intermediate time scale, phase separation occurs and a sharp interface appears which evolves according to a Stefan free boundary problem (and the density function eventually relaxes to a characteristic function - metastable steady state for the original problem). At a larger time scale the attractive forces lead to surface tension phenomena and the evolution of the sharp interface can be described by a Hele-Shaw free boundary problem with surface tension. At that same time scale, we will also discuss the emergence of contact angle conditions for problems set in bounded domains.

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 obtained as gradient flows of interaction-like energies approximating the internal energy. We construct weak solutions as the limit of a (sub)sequence of weak measure solutions by using the Jordan-Kinderlehrer-Otto scheme from the context of $2$-Wasserstein gradient flows. Our strategy allows to cover the porous medium equation, for the general slow diffusion case, extending previous results in the literature. As a byproduct of our analysis, we provide a qualitative particle approximation., Comment: 39 pages, revised with more precise scaling for modulus of convexity in Section 6

We study a regularized version of the Landau equation, which was recently introduced in~\cite{CHWW20} to numerically approximate the Landau equation with good accuracy at reasonable computational cost. We develop the existence and uniqueness theory for weak solutions, and we reinforce the numerical findings in~\cite{CHWW20} by rigorously proving the validity of particle approximations to the regularized Landau equation., Comment: 25 pages (+10 for appendix and references)

We revisit the grazing collision limit connecting the Boltzmann equation to the Landau(-Fokker-Planck) equation from their recent reinterpretations as gradient flows. Our results are in the same spirit as the $\Gamma$-convergence of gradient flows technique introduced by Sandier and Serfaty. In this setting, the grazing collision limit reduces to showing the lower semi-continuous convergence of the Boltzmann entropy-dissipation to the Landau entropy-dissipation., Comment: 41.5 pages without references, 4 figures. Comparisons with other weak solutions have been expanded. Typos fixed and more clarifications added. Accepted version. Comments welcome!

We propose a gradient flow perspective to the spatially homogeneous Landau equation for soft potentials. We construct a tailored metric on the space of probability measures based on the entropy dissipation of the Landau equation. Under this metric, the Landau equation can be characterized as the gradient flow of the Boltzmann entropy. In particular, we characterize the dynamics of the PDE through a functional inequality which is usually referred as the Energy Dissipation Inequality. Furthermore, analogous to the optimal transportation setting, we show that this interpretation can be used in a minimizing movement scheme to construct solutions to a regularized Landau equation., Comment: 46 pages, accepted for publication

We propose a novel deterministic particle method to numerically approximate the Landau equation for plasmas. Based on a new variational formulation in terms of gradient flows of the Landau equation, we regularize the collision operator to make sense of the particle solutions. These particle solutions solve a large coupled ODE system that retains all the important properties of the Landau operator, namely the conservation of mass, momentum and energy, and the decay of entropy. We illustrate our new method by showing its performance in several test cases including the physically relevant case of the Coulomb interaction. The comparison to the exact solution and the spectral method is strikingly good maintaining 2nd order accuracy. Moreover, an efficient implementation of the method via the treecode is explored. This gives a proof of concept for the practical use of our method when coupled with the classical PIC method for the Vlasov equation., Comment: 27 pages, 14 figures, debloated some figures, improved explanations in sections 2, 3, and 4