LOCAL WELL-POSEDNESS FOR THE BOLTZMANN EQUATION WITH VERY SOFT POTENTIAL AND POLYNOMIALLY DECAYING INITIAL DATA

In this paper, we address the local well-posedness of the spatially inhomogeneous noncutoff Boltzmann equation when the initial data decays polynomially in the velocity variable. We consider the case of very soft potentials γ + 2s < 0. Our main result completes the picture for local well-posedness in this decay class by removing the restriction γ + 2s > - 3/2 of previous works. Our approach is entirely based on the Carleman decomposition of the collision operator into a lower order term and an integro-differential operator similar to the fractional Laplacian. Interestingly, this yields a very short proof of local well-posedness when γ ε ( - 3, 0] and s ε (0, 1/2) in a weighted C¹ space that we include as well.