Mathematics – Analysis of PDEs
Scientific paper
2010-02-18
Mathematics
Analysis of PDEs
This file has not changed, but this work has been combined with (arXiv:0912.0888v1), simplified and extended into a new prepri
Scientific paper
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).
Gressman Philip T.
Strain Robert M.
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