R. Marra

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.

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 statistical mechanics of the gauge invariant Ising model

Some results on the phase structure of the gauge invariant Ising model are derived by using convergent expansions.

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...

Hydrodynamic limits of the vlasov equation

We study the Vlasov equation for repulsive forces in the hydrodynamic regime. For initial distributions at zero temperature the limit equations turn out to be the compressible and incompressible Euler equations under suitable space-time scalings.

Solutions to the Boltzmann Equation in the Boussinesq Regime

AbstractWe consider a gas in a horizontal slab in which the top and bottom walls are kept at different temperatures. The system is described by the Boltzmann equation (BE) with Maxwellian boundary conditions specifying the wall temperatures. We study the behavior of the system when the Knudsen number ∈ is small and the temperature difference between the walls as well as the velocity field is of order ∈, while the gravitational force is of order ∈2. We prove that there exists a solution to the BE for % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaerbmv% 3yPrwyGm0BUn3BSvgaiuaacaWF1oWaaeWaaeaacaaIWaGaaiilamaa% naaabaGaamiDaaaaaiaawIcacaGLPaaaaaa!4184! which is near a global Maxwellian, and whose moments are close, up to order ∈2, to the density, velocity and temperature obtained from the smooth solution of the Oberbeck–Boussinesq equations assumed to exist for % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamrr1n% gBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8hzIq4aa0aa% aeaacaWG0baaaaaa!4322! .