R. Esposito

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.

Hydrodynamic limit of the stationary Boltzmann equation in a slab

We study the stationary solution of the Boltzmann equation in a slab with a constant external force parallel to the boundary and complete accommodation condition on the walls at a specified temperature. We prove that when the force is sufficiently small there exists a solution which converges, in the hydrodynamic limit, to a local Maxwellian with parameters given by the stationary solution of the corresponding compressible Navier-Stokes equations with no-slip boundary conditions. Corrections to this Maxwellian are obtained in powers of the Knudsen number with a controlled remainder.

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.

The Navier-Stokes limit of stationary solutions of the nonlinear Boltzmann equation

We consider the flow of a gas in a channel whose walls are kept at fixed (different) temperatures. There is a constant external force parallel to the boundaries which may themselves also be moving. The system is described by the stationary Boltzmann equation to which are added Maxwellian boundary conditions with unit accommodation coefficient. We prove that when the temperature gap, the relative velocity of the planes, and the force are all sufficiently small, there is a solution which converges, in the hydrodynamic limit, to a local Maxwellian with parameters given by the stationary solution of the corresponding compressible Navier-Stokes equations with no-slip voundary conditions. Corrections to this Maxwellian are obtained in powers of the Knudsen number with a controlled remainder.

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.

Binary Fluids with Long Range Segregating Interaction. I: Derivation of Kinetic and Hydrodynamic Equations

We study the evolution of a two component fluid consisting of “blue” and “red” particles which interact via strong short range (hard core) and weak long range pair potentials. At low temperatures the equilibrium state of the system is one in which there are two coexisting phases. Under suitable choices of space-time scalings and system parameters we first obtain (formally) a mesoscopic kinetic Vlasov–Boltzmann equation for the one particle position and velocity distribution functions, appropriate for a description of the phase segregation kinetics in this system. Further scalings then yield Vlasov–Euler and incompressible Vlasov–Navier–Stokes equations. We also obtain, via the usual truncation of the Chapman–Enskog expansion, compressible Vlasov–Navier–Stokes equations.

Navier-Stokes equations for stochastic particle systems on the lattice

We introduce a class of stochastic models of particles on the cubic lattice ℤd with velocities and study the hydrodynamical limit on the diffusive spacetime scale. Assuming special initial conditions corresponding to the incompressible regime, we prove that in dimensiond≧3 there is a law of large numbers for the empirical density and the rescaled empirical velocity field. Moreover the limit fields satisfy the corresponding incompressible Navier-Stokes equations, with viscosity matrices characterized by a variational formula, formally equivalent to the Green-Kubo formula.

On the Derivation of Hydrodynamics from the Boltzmann Equation

We review the main ideas on the derivation of hydrodynamical equations from microscopic models. The Boltzmann equation, which is a good approximation for the evolution of rare gases, provides an useful tool to test these ideas in mathematically controllable situations such as the Euler and incompressible Navier-Stokes limits, which we describe in some detail. We also discuss the heuristics and some few rigorous results available for stochastic particle systems.

Free energy minimizers for a two-species model with segregation and liquid-vapour transition

We study the coexistence of phases in a two-species model whose free energy is given by the scaling limit of a system with long range interactions (Kac potentials) that are attractive between particles of the same species and repulsive between different species.

The milne problem with a force term

We study the stationary half-space linearized Boltzmann equation with a force term decaying to zero at infinity. We extend to this case the results of Bardos, Cafiisch and Nicolaenko for a gas of h...