The Cauchy problem for the inhomogeneous\\ non-cutoff Kac equation in critical Besov space

Hongmei Cao , Hao-Guang Li (1) , Chao-Jiang Xu (2) , Jiang Xu (3)

1 School of mathematics and statistics, South-central university for nationalities, Wuhan
2 LMRS - Laboratoire de Mathématiques Raphaël Salem
3 CRPP - Centre de Recherche Paul Pascal

In this work, we investigate the Cauchy problem for the spatially inhomogeneous non-cutoff Kac equation. If the initial datum belongs to the spatially critical Besov space, we can prove the well-posedness of weak solution under a perturbation framework. Furthermore, it is shown that the solution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable. In comparison with the recent result in [18], the Gelfand-Shilov regularity index is improved to be optimal. To the best of our knowledge, our work is the first one that exhibits smoothing effect for the kinetic equation in Besov spaces.

Domaines: Equations aux dérivées partielles [math.AP]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. The Cauchy problem for the inhomogeneous\\ non-cutoff Kac equation in critical Besov space. 2019. ⟨hal-02021526⟩