偏微分方程研讨班：Analytic smoothing effect of a class of ultra-parabolic equations

中科院数学与系统科学研究院

数学研究所

学术报告

偏微分方程研讨班

报告人：Chaojiang Xu (南京航空航天大学)

题  目：Analytic smoothing effect of a class of ultra-parabolic equations

时  间：2023.11.24（星期五）09:00-10:00

地  点：思源楼S813

摘 要：We study a class of ultra-parabolic equations, it is a high order degenerate parabolic operators of Hormander type, so it is strongly degenerate, but we prove that this class operators possesses the analytic smoothing effect of Cauchy problem, that means, for the initial datum belongs to $L^2$, we prove that the solution is analytic for all spatial variables when $t>$. This class operators contains many kinetic operators such as, Kolmogorov-Fakker-Planck operators, Landau operators and non-cuttoff Boltzmann operators.  

To overcome the degeneracy in the spatial variable, a family of well-chosen vector fields with time-dependent coefficients will play a crucial role, and the analytic regularization effect of weak solutions relies on a quantitative estimate on directional derivatives in these vector fields.