On Hydrodynamic Implosions and the Landau-Coulomb Equation

TL;DR

This paper establishes a new continuation criterion for the Landau equation with Coulomb potential, ruling out certain blow-up scenarios and suggesting limitations on hydrodynamic limit approaches to singularity formation.

Contribution

It introduces a novel continuation criterion based on a quantity not controlling mass density, and rules out tail fattening and certain blow-up rates for the Landau equation.

Findings

Ruling out tail fattening implosion scenario.

Eliminating all Type II self-similar blow-up rates slower than Type I.

Implication that hydrodynamic limits cannot produce singular solutions.

Abstract

We study the inhomogeneous Landau equation with Coulomb potential and derive a new continuation criterion: a smooth solution can be uniquely continued for as long as it remains bounded. This provides, to our knowledge, the first continuation criterion based on a quantity not controlling the mass density. Consequently, we are able to rule out a potential singularity formation scenario known as tail fattening, in which an implosion occurs due to the loss of decay at large $v$. More generally, we are able to rule out all Type II approximately self-similar blow-up rates that are slower than the Type I blow-up rate, without any assumption of decay on the inner profile, complementing existing Type I blow-up analysis in the literature. Heuristically, this suggests that it should be impossible to directly use the hydrodynamic limit connection with the 3D compressible Euler equations to construct a singular solution to the Landau equation with Coulomb potential. Such a potential implosion scenario -- based on either an isentropic or nonisentropic implosion for the 3D Euler equations -- would naturally result in a slow Type II approximately self-similar blow-up scenario, falling well within the range our theorem. This preprint has been subsumed by a more recent work by the authors and Luis Silvestre titled ``Pointwise bounds and obstructions to blowup for the Landau and Boltzmann equations,'' arXiv:2605.20426. This manuscript will remain a permanent preprint; all references should be directed to the more recent work.