This is a brief announcement of our recent proof of global existence and rapid decay to equilibrium of classical solutions to the Boltzmann equation without any angular cutoff, that is, for long-range interactions. We consider perturbations of the Maxwellian equilibrium states and include the physical cross-sections arising from an inverse-power intermolecular potential r -(p-1)
with p > 2, and more generally. We present here a mathematical framework for unique global in time solutions for all of these potentials. We consider it remarkable that this equation, derived by Boltzmann in 1872 and Maxwell in 1867, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world . Our methods provide a new understanding of the effects due to grazing collisions.
http://www.pnas.org/content/107/13/5744.abstract?sid=9d675b61-b4d1-4bc0-88d1-b450d0b4b5ca
Philip T. Gressman and Robert M. Strain
University of Pennsylvania, Department of Mathematics, Philadelphia, PA 19104-6395
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