De Giorgi argument for weighted L2∩ L∞ solutions to the non-cutoff Boltzmann equation

Abstract

This paper gives an affirmative answer to the question of the global existence of Boltzmann equations without angular cutoff in the L∞-setting. In particular, we show that when the initial data is close to equilibrium and the perturbation is small in L2∩ L∞ with a polynomial decay tail, the Boltzmann equation has a unique global solution in the weighted L2∩ L∞-space. In order to overcome the difficulties arising from the singular cross-section and the low regularity, a De Giorgi type argument is crafted in the kinetic context with the help of the averaging lemma. More specifically, we use a strong averaging lemma to obtain suitable Lp-estimates for level-set functions. These estimates are crucial for constructing an appropriate energy functional to carry out the De Giorgi argument. Similar as in Alonso et al. (Rev Mat Iberoam, 2020), we extend local solutions to global ones by using the spectral gap of the linearised Boltzmann operator. The convergence to the equilibrium state is then obtained as a byproduct with relaxations shown in both L2 and L∞-spaces.