  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).
