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 non-cutoff Boltzmann equation when the initial data decays polynomially in the velocity variable. We consider the case of very soft potentials $\gamma + 2s < 0$. Our main result completes the picture for local well-posedness in this decay class by removing the restriction $\gamma + 2s > -3/2$ 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 $\gamma \in (-3,0]$ and $s \in (0,1/2)$ in a weighted $C^1$ space that we include as well.