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

Abstract

We study the slightly perturbed homogeneous Landau equation \[ \partial_t f = a_{ij}(f) \cdot \partial_{ij} f + αc(f) f, \quad c(f) = - \partial_{ij} a_{ij}(f), \] 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 $f_0\ge 0$, 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.

35 pages