Gelfand-Shilov smoothing effect for the Cauchy problem of the non-cutoff spatially homogeneous Boltzmann equation

Leo Glangetas (1) , Hao-Guang Li (2) , Chao-Jiang Xu (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem
2 School of Mathematics and Statistics [Wuhan]

In this work, we prove the Gelfand-Shilov smoothing effect of the Cauchy problem for the spatially homogeneous non-cutoff Boltzamnn equation with Maxwellian molecules. The smoothing properties is same as the Cauchy problem defined by the evolu-tion equation associated to a fractional harmonic oscillator. We also construct the solution of the Boltzmann equation by solving an infinite systems of ordinary differential equations. The key tools is the spectral decomposition of linear and non-linear Boltzmann operators,

Mots clés: Boltzmann equation; spectral decomposition; Gelfand-Shilov class

Domaines: Equations aux dérivées partielles [math.AP]

Leo Glangetas, Hao-Guang Li, Chao-Jiang Xu. Gelfand-Shilov smoothing effect for the Cauchy problem of the non-cutoff spatially homogeneous Boltzmann equation. 2014. ⟨hal-01088989v1⟩