Abstract:
We introduce a class of $L^p$ solutions to the Boltzmann equation with an $varepsilon$-truncated, pseudo-Maxwell molecule collision kernel which, as $varepsilon$ vanishes, approach a weak solution of the Landau equation in its most singular form - the Coulomb potential. Such a pseudo-Maxwellian kernel was created in a paper of Bobylev and Potapenko. It is an angle concentrated measure over the unit sphere, with mass near the direction of the relative post-collisional velocity of two interacting particles. This type of kernel has a big advantage because its angular average over the sphere is finite and has no singularities that would otherwise be present in the traditional kernel. This allows us to split the collision operator into its gain and loss parts, at which point we can perform analyses and $L^p$ estimates, motivated by the works of Desvillettes, Lions and Villani as well as Alonso, Gamba and Taskovic. |