On Hydrodynamic Implosions and the Landau-Coulomb Equation
William Golding, Christopher Henderson
TL;DR
This work analyzes the inhomogeneous Landau equation with Coulomb potential and proves a sharp continuation criterion: a smooth solution can be uniquely extended as long as the time-integrated $L^\infty_{x,v}$ bound remains finite, a criterion that does not control mass density and thus rules out tail fattening. Central to the result is a robust a priori estimate: for non-integer $m\in(2,5)$, the weighted function $g=\langle v\rangle^m f$ satisfies $\|f(t)\|_{L^\infty_m} \le e^{K\int_0^t \|f(s)\|_{L^\infty} ds}\|f_{\rm in}\|_{L^\infty_m}$, enabling continuation under the prescribed bound. The authors also derive precise weighted bounds for the diffusion coefficient $A[f]$ and perform a detailed main computation to obtain the pivotal upper bound; these results collectively rule out implosion scenarios driven by tail fattening and substantially constrain almost all Type II self-similar blow-up rates. The work further connects these kinetic-continuation insights to hydrodynamic limits, showing that hydrodynamic (Euler) implosion mechanisms are incompatible with building singular Landau solutions via Maxwellian leading terms, thereby clarifying the landscape of possible singular behaviors for the Coulomb Landau equation.
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 velocities. More generally, we are able to rule out almost all Type II approximately self-similar blow-up rates, 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 non-isentropic implosion for the 3D Euler equations -- would naturally result in a Type II approximately self-similar blow-up scenario, falling well within the range our theorem.