Velocity Averaging Lemmas in Hyperbolic Sobolev Spaces for the Kinetic Transport Equation with Velocity Field on the Sphere

Abstract

We show how the methods of [6–8] can be used to prove velocity averaging lemmas in hyperbolic Sobolev spaces for the kinetic transport equation \(\partial_{t}f + \frac {v} {|v|} \cdot \nabla_{x}f = g_{0} + \nabla_\nu \cdot g_{1}\). Here v is allowed to vary in the whole space \({\mathbb{R}}^{d}\) and the velocity field \(a(v) = \frac {v} {|v|}\) lies on the unit sphere. We work in dimensions \(d \geq 3\) and, in contrast with [6, 8], we allow right-hand sides with velocity derivatives in any direction and not necessarily tangential to the sphere.