We present a simple approach to study the one-dimensional pressureless Euler system via adhesion dynamics in the Wasserstein space of probability measures with finite quadratic moments.
Starting from a discrete system of a finite number of "sticky" particles, we obtain new explicit estimates of the solution in terms of the initial mass and momentum and we are able to construct an evolution semigroup in a measure-theoretic phase space, allowing mass distributions with finite quadratic moment and corresponding L^2-velocity fields. We investigate various interesting properties of this semigroup, in particular its link with the gradient flow of the (opposite) squared Wasserstein distance.
Our arguments rely on an equivalent formulation of the evolution as a gradient flow in the convex cone of nondecreasing functions in the Hilbert space L^2(0,1), which corresponds to the Lagrangian system of coordinates given by the canonical monotone rearrangement of the measures.
This article is dedicated to the study of diagonal hyperbolic systems in one space dimension, with cumulative distribution functions, or more generally nonconstant monotonic bounded functions, as initial data. Under a uniform strict hyperbolicity assumption on the characteristic fields, we construct a multitype version of the sticky particle dynamics and obtain existence of global weak solutions by compactness. We then derive a $L^p$ stability estimate on the particle system uniform in the number of particles. This allows to construct nonlinear semigroups solving the system in the sense of Bia…
The sticky particle system is a system of partial differential equations which assert the conservation of mass and momentum of a collection of particles that interact only via inelastic collisions. These equations arise in Zel'dovich's theory for the formation of large scale structures in the universe. We will show that this system of equations has a solution in one spatial dimension for given initial conditions by generating a trajectory mapping in Lagrangian coordinates.
We study collections of point masses which move freely along the real line and stick together when they collide via perfectly inelastic collisions. We quantify the way particles stick together and explain how to associate a probability measure on the space of continuous paths to such a collection of evolving point masses. These observations lead to a new method of designing solutions to the sticky particle system in one spatial dimension which have nonincreasing kinetic energy and satisfy an entropy inequality.
We consider the dynamics of finite systems of point masses which move along the real line. We suppose the particles interact pairwise and undergo perfectly inelastic collisions when they collide. In particular, once particles collide, they remain stuck together thereafter. Our main result is that if the interaction potential is semi-convex, this sticky particle property can quantified and is preserved upon letting the number of particles tend to infinity. This is used to show that solutions of the pressureless Euler equations exist for given initial conditions and satisfy an entropy inequality…
In this paper we consider the multi-dimensional pressureless Euler system and we tackle the problem of existence and uniqueness of sticky particle solutions for general measure-type initial data. Although explicit counterexamples to both existence and uniqueness are known since \cite{Bressan-Nguyen}, the problem of whether one can still find sticky particle solutions for a large set of data and of how one can select them was up to our knowledge still completely open. In this paper we prove that for a comeager set of initial data in the weak topology the pressureless Euler system admits a uniqu…
The paper is concerned with sticky weak solutions to the equations of pressureless gases in two or more space dimensions. Various initial data are constructed, showing that the Cauchy problem can have (i) two distinct sticky solutions, or (ii) no sticky solution, not even locally in time. In both cases the initial density is smooth with compact support, while the initial velocity field is continuous.
We consider compressible pressureless fluid flows in Lagrangian coordinates in one space dimension. We assume that the fluid self-interacts through a force field generated by the fluid itself. We explain how this flow can be described by a differential inclusion on the space of transport maps, in particular when a sticky particle dynamics is assumed. We study a discrete particle approximation and we prove global existence and stability results for solutions of this system. In the particular case of the Euler-Poisson system in the attractive regime our approach yields an explicit representation…
We use a sticky particle method to show global existence of (energy) conservative sticky $N$-peakon solutions to the modified Camassa-Holm equation. A dispersion regularization is provided as a selection principle for the uniqueness of conservative $N$-peakon solutions. The dispersion limit avoids the collision between peakons, and numerical results show that the dispersion limit is exactly the sticky peakons. At last, when the splitting of peakons is allowed, we give an example to show the non-uniqueness of conservative solutions.
We develop a global wellposedness theory for weak solutions to the 1D Euler-alignment system with measure-valued density, bounded velocity, and locally integrable communication protocol. A satisfactory understanding of the low-regularity theory is an issue of pressing interest, as smooth solutions may lose regularity in finite time. However, no such theory currently exists except for a very special class of alignment interactions. We show that the dynamics of the 1D Euler-alignment system can be effectively described by a nonlocal scalar balance law, the entropy conditions of which serves as a…
We investigate ground state configurations for a general finite number $N$ of particles of the Heitmann-Radin sticky disc pair potential model in two dimensions. Exact energy minimizers are shown to exhibit large microscopic fluctuations about the asymptotic Wulff shape which is a regular hexagon: There are arbitrarily large $N$ with ground state configurations deviating from the nearest regular hexagon by a number of $\sim N^{3/4}$ particles. We also prove that for any $N$ and any ground state configuration this deviation is bounded above by $\sim N^{3/4}$. As a consequence we obtain an exact…
We derive an explicit formula for global weak solutions of the one dimensional system of pressure-less Euler-Poisson equations. Our variational formulation is an extension of the well-known formula for entropy solutions of the scalar inviscid Burgers' equation: since the characteristics of the Euler-Poisson equations are parabolas, the representation of their weak solution takes the form of a "quadratic" version of the celebrated Lax-Oleinik variational formula. Three cases are considered. (i) The variational formula recovers the "sticky particle" solution in the attractive case; (ii) It repre…
We study the discretization of generalized Wasserstein distances with nonlinear mobilities on the real line via suitable discrete metrics on the cone of N ordered particles, a setting which naturally appears in the framework of deterministic particle approximation of partial differential equations. In particular, we provide a $Γ$-convergence result for the associated discrete metrics as $N \to \infty$ to the continuous one and discuss applications to the approximation of one-dimensional conservation laws (of gradient flow type) via the so-called generalized minimizing movements, proving a conv…
We propose iterative algorithms to solve adversarial problems in a variety of supervised learning settings of interest. Our algorithms, which can be interpreted as suitable ascent-descent dynamics in Wasserstein spaces, take the form of a system of interacting particles. These interacting particle dynamics are shown to converge toward appropriate mean-field limit equations in certain large number of particles regimes. In turn, we prove that, under certain regularity assumptions, these mean-field equations converge, in the large time limit, toward approximate Nash equilibria of the original adv…
These notes are devoted to a summary on the mean-field limit of large ensembles of interacting particles with applications in swarming models. We first make a summary of the kinetic models derived as continuum versions of second order models for swarming. We focus on the question of passing from the discrete to the continuum model in the Dobrushin framework. We show how to use related techniques from fluid mechanics equations applied to first order models for swarming, also called the aggregation equation. We give qualitative bounds on the approximation of initial data by particles to obtain t…
We investigate the sedimentation of identical inertialess spherical particles in a Stokes fluid in the limit of many small particles. It is known that the presence of the particles leads to an increase of the effective viscosity of the suspension. By Einstein's formula this effect is of the order of the particle volume fraction $ϕ$. The disturbance of the fluid flow responsible for this increase of viscosity is very singular (like $|x|^{-2}$). Nevertheless, for well-prepared initial configurations and $ϕ\to 0$, we show that the microscopic dynamics is approximated to order $ϕ^2 |\log ϕ|$ by a …
On the asymptotic behavior of the Repulsive Pressureless Euler-Poisson System
Columbia University（哥伦比亚大学）；Wayne State University（韦恩州立大学）；Worcester Polytechnic Institute（伍斯特理工学院）；University of Minnesota（明尼苏达大学）；University of Connecticut（康涅狄格大学）；West Virginia University（西弗吉尼亚大学）
We study the connection between a system of many independent Brownian particles on one hand and the deterministic diffusion equation on the other. For a fixed time step $h>0$, a large-deviations rate functional $J_h$ characterizes the behaviour of the particle system at $t=h$ in terms of the initial distribution at $t=0$. For the diffusion equation, a single step in the time-discretized entropy-Wasserstein gradient flow is characterized by the minimization of a functional $K_h$. We establish a new connection between these systems by proving that $J_h$ and $K_h$ are equal up to second order in …
This article is a continuation of our first work \cite{chaudruraynal:frikha}. We here establish some new quantitative estimates for propagation of chaos of non-linear stochastic differential equations in the sense of McKean-Vlasov. We obtain explicit error estimates, at the level of the trajectories, at the level of the semi-group and at the level of the densities, for the mean-field approximation by systems of interacting particles under mild regularity assumptions on the coefficients. A first order expansion for the difference between the densities of one particle and its mean-field limit is…
Sticky particle solutions to the one-dimensional pressureless gas dynamics equations can be constructed by a suitable metric projection onto the cone of monotone maps, as was shown in recent work by Natile and Savaré. Their proof uses a discrete particle approximation and stability properties for first order differential inclusions. Here we give a more direct proof that relies on a result by Haraux on the differentiability of metric projections. We apply the same method also to the one-dimensional Euler-Poisson system, obtaining a new proof for the global existence of weak solutions.
We study the large deviation rate functional for the empirical distribution of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer that this functional is asymptotically equivalent (in the sense of $Γ$-convergence) to the Jordan--Kinderlehrer--Otto functional arising in the Wasserstein gradient flow structure of the Fokker--Planck equation. In higher dimensions, part of this statement (the lower bound) has been recently proved by Duong, Laschos and Renger, but the upper bound remained open, since the proof in \cite{DLR2013} relies …
