Abstract
In this paper, we prove estimates of the smoothing property of the gain part of the Boltzmann collision operator for the full kernel of the hard sphere model in R3 which are an improvement over estimates by Lions [J. Math. Kyoto Univ., 34 (1994), pp. 391-427, pp. 429-461] and Wennberg [Comm. Partial Differential Equations, 19 (1994), pp. 2057-2074]. In earlier works for different collision models, this type of smoothing estimate has always come with the compactness assumption [P.-L. Lions, J. Math. Kyoto Univ., 34 (1994), pp. 391-427, 429-461] for both variables of the collision kernel. Now we show that the decaying condition in the angular component which is caused by non-head-on collisions for the hard sphere model can replace the compactness assumption. Namely, no artificial truncation of the angular component at the boundary points is needed to prove the regularizing effect. Also the compactness assumption in relative velocity can be relaxed. For large relative velocity the smoothing estimate can hold in proper weighted function space. For small relative velocity, we can only prove a weaker smoothing estimate. © 2012 Society for Industrial and Applied Mathematics.