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

TL;DR

This paper demonstrates that for the slightly perturbed homogeneous Landau equation with very soft potentials, solutions can develop nearly self-similar blowup in finite time, contrasting the global well-posedness at the unperturbed case.

Contribution

The paper establishes finite time nearly self-similar blowup for the perturbed Landau equation with very soft potentials, extending previous frameworks to this kinetic model.

Findings

Finite time blowup occurs for lpha > 1 close to 1.

Blowup can be both radially and non-radially symmetric.

Results highlight potential singularity formation in inhomogeneous Landau equations.

Abstract

We study the slightly perturbed homogeneous Landau equation \[ \partial_t f = a_{ij}(f) \cdot \partial_{ij} f + \alpha c(f) f, \quad c(f) = - \partial_{ij} a_{ij}(f), \] 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 initial data $f_0 \geq 0$, which can be both radially symmetric or non-radially symmetric. The blowup results are sharp as the homogeneous Landau equation $(\alpha=1)$ is globally well-posed, which was established recently by Guillen and Silvestre. To prove the blowup results, we build on our previous framework \cite{chen2020slightly,chen2021regularity} 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.