Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0902.4373

A Wasserstein approach to the one-dimensional sticky particle system

Luca Natile·Giuseppe Savaré
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.APmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Luca Natile, Giuseppe Savaré. 2009-04-16. A Wasserstein approach to the one-dimensional sticky particle system. https://doi.org/10.1137/090750809

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Blow-Up Dynamics for the $L^2$ critical case of the $2$D Zakharov-Kuznetsov equation

We study blow-up dynamics for the $L^2$-critical cubic Zakharov--Kuznetsov equation in two dimensions, \[ \partial_tu+\partial_{x_1}(Δu+u^3)=0 \qquad\text{on }\mathbb R^2. \] For a class of localized $H^1$ perturbations of the ground state $Q$, we establish a trichotomy near the soliton manifold: exit from a small $L^2$-tube, global asymptotic stability, or finite-time blow-up. In the stable blow-up regime, the solution concentrates a single bubble and \[ λ(t)\sim \ell_0(T-t)^{1/(3-c)}, \] where $\ell_0>0$ depends on the initial datum and $c\in(1,2)$ is an explicit constant determined by the transverse tail of the first-order approximate profile. Consequently, \[ \|\nabla u(t)\|_{L^2} \sim \frac{\|\nabla Q\|_{L^2}} {\ell_0(T-t)^{1/(3-c)}}. \] After subtraction of the concentrating soliton, the radiation converges strongly in $L^p(\mathbb R^2)$ for every $2\leq p<\infty$ to a common nonzero profile $u^*$, while \[ u^*\notin H^s(\mathbb R^2) \qquad\text{for every }s\geq\frac c2. \] The stable blow-up branch is open in the relative $H^1$ topology of the localized class. Finally, every non-soliton datum in this class with non-positive energy blows up in finite time. Interval-arithmetic computer-assisted proofs certify the numerical inputs to the virial coercivity argument. They also yield a rigorous enclosure of $c$, justifying the polynomial moment of order $21$ imposed on the initial data.

math.AP↗
arXiv2026

Propagation of wave packets close to conical intersections

In this paper, we study the propagation of wave packets close to conical intersections with respect to a system of two Schr{ö}dinger equations presenting a codimension 2 crossing. We focus on the dynamics that occur when the wave packets pass through an area close to the crossing, and our main results provide an explicit formula for the outgoing wave packet in terms of the incoming one, with a complete description of its phase and of the classical trajectories it follows, including a drift.

math.AP↗
arXiv2026

A Volterra Calculus for Lie Groupoids

A pseudodifferential Volterra calculus for inverting parabolic differential equations on Lie groupoids is introduced. This enables the study of fundamental solutions of various cases of heat flows on singular manifolds with corners with non-resonant boundary indicial symbols, such as the $b$-manifolds, as well as other geometric bisection covariant heat flows. We also establish the short time asymptotic expansion for the heat kernel of a positive, elliptic differential operator on a Lie groupoid that acts on suitable Sobolev Hilbert modules and is positive definite with respect to the appropriate $L^2$ inner product.

math.AP↗