Mario Pulvirenti

Macroscopic Description of Microscopically Strongly Inhomogenous Systems: A Mathematical Basis for the Synthesis of Higher Gradients Metamaterials

We consider the time evolution of a one dimensional n-gradient continuum. Our aim is to construct and analyze discrete approximations in terms of physically realizable mechanical systems, referred to as microscopic because they are living on a smaller space scale. We validate our construction by proving a convergence theorem of the microscopic system to the given continuum, as the scale parameter goes to zero.

A Non-Maxwellian Steady Distribution for One-Dimensional Granular Media

We consider a nonlinear Fokker–Planck equation for a one-dimensional granular medium. This is a kinetic approximation of a system of nearly elastic particles in a thermal bath. We prove that homogeneous solutions tend asymptotically in time toward a unique non-Maxwellian stationary distribution.

The Boltzmann equation for weakly inhomogeneous data

We solve the initial value problem associated to the nonlinear Boltzmann equation in the case in which the initial distribution has sufficiently small spatial gradients.

On the validity of the Boltzmann equation for short range potentials

We consider a classical system of point particles interacting by means of a short range potential. We prove that, in the low-density (Boltzmann–Grad) limit, the system behaves, for short times, as predicted by the associated Boltzmann equation. This is a revisitation and an extension of the thesis of King [9] (that appeared after the well-known result of Lanford [10] for hard spheres) and of a recent paper by Gallagher et al. [5]. Our analysis applies to any stable and smooth potential. In the case of repulsive potentials (with no attractive parts), we estimate explicitly the rate of convergence.

SOME CONSIDERATIONS ON THE DERIVATION OF THE NONLINEAR QUANTUM BOLTZMANN EQUATION.

In this paper we analyze a system of Nidentical quantum particles in a weak-coupling regime. The time evolution of the Wigner transform of the one-particle reduced density matrix is represented by means of a perturbative series. The expansion is obtained upon iterating the Duhamel formula. For short times, we rigorously prove that a subseries of the latter, converges to the solution of the Boltzmann equation which is physically relevant in the context. In particular, we recover the transition rate as it is predicted by Fermis Golden Rule. However, we are not able to prove that the quantity neglected while retaining a subseries of the complete original perturbative expansion, indeed vanishes in the limit: we only give plausibility arguments in this direction. The present study holds in any space dimension d≥2.

Euler evolution for singular initial data and vortex theory

We study the evolution of a two dimensional, incompressible, ideal fluid in a case in which the vorticity is concentrated in small, disjoint regions of the physical space. We prove, for short times, a connection between this evolution and the vortex model.

A stochastic system of particles modelling the Euler equations

We consider a system of N spheres interacting through elastic collisions at a stochastic distance. In the limit N → ∞, for a suitable rescaling of the interaction parameters, we prove that the one-particle distribution function converges to a local Maxwellian, whose gross density, velocity, and temperature satisfy the Euler equation.

ON THE WEAK-COUPLING LIMIT FOR BOSONS AND FERMIONS

In this paper we consider a large system of bosons or fermions. We start with an initial datum which is compatible with the Bose–Einstein, respectively Fermi–Dirac, statistics. We let the system of interacting particles evolve in a weak-coupling regime. We show that, in the limit, and up to the second order in the potential, the perturbative expansion expressing the value of the one-particle Wigner function at time t, agrees with the analogous expansion for the solution to the Uehling–Uhlenbeck equation. This paper follows the same spirit as the companion work,2 where the authors investigated the weak-coupling limit for particles obeying the Maxwell–Boltzmann statistics: here, they proved a (much stronger) convergence result towards the solution of the Boltzmann equation.

Vortices and localization in Euler flows

We study the time evolution of a non-viscous incompressible two-dimensional fluid when the initial vorticity is concentrated inN small disjoint regions of diameter ε. We prove that the time evolved vorticity is also concentrated inN regions of diameterd, vanishing as ε→0. As a consequence we give a rigorous proof of the validity of the point vortex system. The same problem is examined in the context of the vortex-wave system.

Some considerations on the nonlinear stability of stationary planar Euler flows

We give sufficient conditions for the nonlinear stability of possibly nonsmooth stationary solutions of the two-dimensional Euler equation in symmetric bounded domains. We use, as Lyapunov functions, first integrals due to the symmetry of the problem. Moreover, we investigate the stability of smooth solutions under perturbations of the boundary. The last result is based on a generalization of the well known Arnold approach.