On Semi-classical Limit of Spatially Homogeneous Quantum Boltzmann Equation: Weak Convergence

On the Grazing Collision Limit for the Spatially Homogeneous Boltzmann Equation

Global Weak Solutions to the Compressible Quantum Navier-stokes Equation and Its Semi-classical Limit

This paper is dedicated to the construction of global weak solutions to the quantum Navier-Stokes equation, for any initial value with bounded energy and entropy. The construction is uniform with respect to the Planck constant. This allows to perform the semi-classical limit to the associated compressible Navier-Stokes equation. One of the difficulty of the problem is to deal with the degenerat...

On Stability and Strong Convergence for the Spatially Homogeneous Boltzmann Equation for Fermi-dirac Particles

ABSTRACT. The paper considers the stability and strong convergence to equilibrium of solutions to the spatially homogeneous Boltzmann equation for Fermi-Dirac particles. Under the usual cut-off condition on the collision kernel, we prove a strong stability in L-topology at any finite time interval, and, for hard and Maxwellian potentials, we prove that the solutions converge strongly in L to eq...

Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials

For the spatially homogeneous Boltzmann equation with hard potentials and Grad’s cutoff (e.g. hard spheres), we give quantitative estimates of exponential convergence to equilibrium, and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation, on which we provide a lower bound. Our approach is based on establishing spectral gaplike estimates valid ...

Upper Maxwellian Bounds for the Spatially Homogeneous Boltzmann Equation

For the spatially homogeneous Boltzmann equation with cutoff hard potentials it is shown that solutions remain bounded from above, uniformly in time, by a Maxwellian distribution, provided the initial data have a Maxwellian upper bound. The main technique is based on a comparison principle that uses a certain dissipative property of the linear Boltzmann equation. Implications of the technique t...