Global Classical Solutions of the Boltzmann Equation with Long-Range Interactions and Soft Potentials

Abstract

In this work we prove global stability for the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse power intermolecular potentials, $r^{-(p-1)}$ with $p>2$. This completes the work which we began in (arXiv:0912.0888v1). We more generally cover collision kernels with parameters $s\in (0,1)$ and $\gamma$ satisfying $\gamma >-(n-2)-2s$ in arbitrary dimensions ${\mathbb{T}}^n\times {\mathbb{R}}^n$ with $n\ge 2$. Moreover, we prove rapid convergence as predicted by the Boltzmann H-Theorem. When $\gamma +2s\ge 0$, we have exponential time decay to the Maxwellian equilibrium states. When $\gamma +2s<0$, our solutions decay polynomially fast in time with any rate. These results are constructive. Additionally, we prove sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when $\gamma +2s\ge 0$, as conjectured in Mouhot-Strain (2007).

This file has not changed, but this work has been combined with (arXiv:0912.0888v1), simplified and extended into a new preprint, please see the updated version: arXiv:1011.5441v1

References in corpus (4)

• Global Classical Solutions of the Boltzmann Equation without Angular Cut-off
• Global Strong Solutions of the Boltzmann Equation without Angular Cut-off
• Global Classical Solutions of the Boltzmann Equation with Long-Range Interactions and Soft Potentials
• On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity

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