Cauchy problem for the spatially homogeneous landau equation with shubin class initial datum and gelfand-shilov smoothing effect

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

1 School of mathematics and statistics, South-central university for nationalities, Wuhan
2 LMRS - Laboratoire de Mathématiques Raphaël Salem

In this work, we study the nonlinear spatially homogeneous Lan-dau equation with Maxwellian molecules, by using the spectral analysis, we show that the non linear Landau operators is almost linear, and we prove the existence of weak solution for the Cauchy problem with the initial datum belonging to Shubin space of negative index which conatins the probability measures. Based on this spectral decomposition, we prove also that the Cauchy problem enjoys Gelfand-Shilov smoothing effect, meaning that the weak solution of the Cauchy problem with Shubin class initial datum is ultra-analytics and exponential decay for any positive time.

Mots clés: Spatially homogeneous Landau equation; spectral decomposition; ultra-analytic smoothing effect; Shubin space; Gelfand-Shilov space

Domaines: Equations aux dérivées partielles [math.AP]

Hao-Guang Li, Chao-Jiang Xu. Cauchy problem for the spatially homogeneous landau equation with shubin class initial datum and gelfand-shilov smoothing effect. SIAM Journal on Mathematical Analysis, 2019, 51 (1), pp.532-564. ⟨hal-01576108⟩