Abstract:
We consider the space homogeneous Boltzmann equation with an $eps$-parameter $|u|$ bounded collision kernel, introduced by Bobylev and Potapenko in 2013, that enables us to derive the Landau equation as the scattering angle vanishes when $eps$ goes to zero. We show that solutions of such a Boltzmann problem are in $L^p(R^3)$ uniformly in time $t$ and in $eps$, by using an analog of a result from 2010 of Young's inequality for the collision operator with such bounded cross section (Alonso, Carneiro, Gamba). In addition, we obtain gain of integrability estimates for this problem by constructing a nonlinear ordinary differential Bernoulli type inequality for $|f_{eps}|(t)$. The case $p = 3$ is of particular interest, since it was shown in 2015 by L. Desvillettes that H-solutions of the Landau equation are in fact classical weak solutions which lie in $L^3_{-3}$ whenever $f_0 in L^1_2cap Llog L(R^3).$ We will discuss the global in time weak convergence of the Boltzmann solutions sequence $f_{eps}(v,t)$ in $L^3_{-3}$ to a global Desvillettes type solution of the Landau equation. |