### Abstract or Additional Information

In this colloquium talk we explain several recent results surrounding global stability problem for the Boltzmann equation (1872) with the physically important collision kernels derived by Maxwell (1867) for the full range of inverse power intermolecular potentials, \(r^{-(p-1)}\) with \(p > 2\) and more generally. This is a problem which had remained open for quite a long time.

Specifically, we now have global solutions that are perturbations of the Maxwellian equilibrium states, and which decay rapidly in time to equilibrium. This proof is facilitated by our new sharp geometric understanding of the diffusive nature of the non cut-off collision operator.

Our methods also provide a new coercive "entropy production" estimate in terms of the same geometric fractional semi-norm as in the linearized context. This semi-norm is sharper than previously known coercive lower bounds, and it seems to be the sharp coercive estimate at the fully non-linear level.

Furthemore, since the work of Ukai-Asano in 1982, it has been a longstanding open problem to determine the optimal large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cut-off assumption. We prove that our solutions converge to the global Maxwellian with the optimal large-time decay rate of \(O(t^{-\frac{n}{2}+\frac{n}{2r}})\) in the \(L^2_v(L^r_x)\)-norm for any \(2\leq r\leq \infty\) in \(n\)-dimensions.

This colloquium talk will include a great deal of historical background and anecdotes. Much of this is joint work with P. Gressman