Smoothing Estimates for Boltzmann Equation with Full-range Interactions: Spatially Inhomogeneous Case

In this paper, we study the regularity of the solution to the Boltzmann equation with full-range interactions but for the spatially inhomogeneous case. Under the initial regularity assumption on the solution itself, we show that the solution will become immediately smooth for all the variables as long as the time is far way from zero. Our strategy relies upon the new upper and lower bounds for the collision operator established in Chen and He (Arch Ration Mech Anal 201(2):501-548, 2011), a hypo-elliptic estimate for the transport equation and the element energy method.