Regularity of solutions for the Boltzmann equation without angular cutoff

Abstract

We prove that classical solution of the spatially inhomogeneous and angular non-cutoff Boltzmann equation is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∞</mml:mo> </mml:msup> </mml:math> with respect to all variables, locally in the space and time variables. The proof relies on a generalized uncertainty principle, some improved upper bound and coercivity estimates on the nonlinear collision operator, and some subtle analysis on the commutators between the collision operators and some appropriately chosen pseudo-differential operators.