A variational representation of weak solutions for the pressureless Euler-Poisson equations

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 represents a repulsive solution which is different than the one obtained by the sticky particle construction; and (iii) the result is further extended to the multi-dimensional Euler-Poisson system with radial symmetry

A Wasserstein approach to the one-dimensional sticky particle system

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.Comment: Added reference

Trasporto ottimo e problemi di evoluzione per sistemi di particelle

L'argomento trattato riguarda un approccio da poter usare in problemi evolutivi in sistemi di particelle. La teoria del trasporto ottimo di massa permette di avere uno spazio metrico nel quale ambientare il problema; il metodo dei flussi gradiente è poi illustrato negli spazi di Hilbert, metodo utile per studiare equazioni differenziali di tipo evolutivo. Il problema che studiamo come esempio è lo Sticky Particle System, un caso speciale delle equazioni di Eulero, per il quale illustriamo un risultato di esistenza di soluzione entropica per una classe vasta di dati inizial