DISSIPATION AND SEMIGROUP ON Hnk : NON-CUTOFF LINEARIZED BOLTZMANN OPERATOR WITH SOFT POTENTIAL

In this paper, we find that the linearized collision operator L of the non-cutoff Boltzmann equation with soft potential generates a strongly continuous semigroup on H-n(k), with k, n is an element of R. In the theory of the Boltzmann equation without angular cutoff, the weighted Sobolev space plays a fundamental role. The proof is based on pseudodifferential calculus, and, in general, for a specific class of Weyl quantization, the L-2 dissipation implies H-n(k), dissipation. This kind of estimate is also known as Garding's inequality.