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

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.