Finite time singularities in the Landau equation with very hard potentials

TL;DR

This paper demonstrates finite time singularities in the inhomogeneous Landau equation with very hard potentials, showing that smooth initial data can lead to blow-up in certain norms while remaining bounded in others.

Contribution

It constructs the first example of a collisional kinetic model that is globally well-posed in the homogeneous case but develops finite time singularities in the inhomogeneous setting.

Findings

$C^{eta}$-norm of the distribution blows up for all $eta>0$

Distribution converges to a local Maxwellian in self-similar variables

Hydrodynamic fields exhibit asymptotic self-similar implosion

Abstract

We consider the inhomogeneous Landau equation with $\gamma \in (\sqrt{3},2]$ and construct smooth, strictly positive initial data that develop a finite time singularity. The $C^{\alpha}$-norm of the distribution function blows up for every $\alpha>0$, whereas its $L^{\infty}$-norm remains uniformly bounded. In self-similar variables, the solution becomes asymptotically hydrodynamic - the distribution function converges to a local Maxwellian, while the hydrodynamic fields develop an asymptotically self-similar implosion whose profile coincides with a smooth imploding profile of the compressible Euler equations. To our knowledge, this provides the first example of a collisional kinetic model which is globally well-posed in the homogeneous setting, but admits finite time singularities for inhomogeneous data.