偏微分方程研讨班：Nearly self-similar blowup of the slightly perturbed homogeneous Landau equation with very soft potentials

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：Jiajie Chen（Courant Institute, New York University）

题  目：Nearly self-similar blowup of the slightly perturbed homogeneous Landau equation with very soft potentials

时  间：2024.04.03（星期三）9:00-10:00

地  点：Zoom ID: 924 888 5804  Passcode: AMSS2022

摘 要：Whether the Landau equation can develop a finite time singularity is an important open problem in kinetic equations. In this talk, we will first discuss several similarities between the Landau equation and some incompressible fluids equations. Then we will focus on the slightly perturbed homogeneous Landau equation with very soft potentials, where we increase the nonlinearity from $ c(f) f$ in the Landau equation to $\alpha c(f) f$ with $\alpha>1$. For $\alpha > 1 $ and close to $1$, we establish finite time nearly self-similar blowup from some smooth non-negative initial data, which can be radially symmetric or non-radially symmetric. The blowup results are sharp as the homogeneous Landau equation $(\alpha=1)$ is globally well-posed, which was recently established by Guillen and Silvestre. The proof builds on our previous framework on sharp blowup results of the De Gregorio model with nearly self-similar singularity to overcome the diffusion. Our results shed light on potential singularity formation in the inhomogeneous setting.