Local well-posedness for the Boltzmann equation with very soft potential and polynomially decaying initial data
We consider 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. It is based on the Carleman decomposition of the collision operator into a lower order term and an integro-differential operator similar to the fractional Laplacian.