A convexity-type invariant for the critical coagulation--fragmentation Hamilton--Jacobi equation

We study the critical coagulation--fragmentation equation with multiplicative coagulation kernel \(a(s,\hat s)=s\hat s\) and constant fragmentation kernel \(b(s,\hat s)=1\).

Under the Bernstein transform, mass-conserving solutions correspond to solutions of a singular Hamilton--Jacobi equation studied by Tran and Van (this http URL Appl.Math.75 (2022), no.6, 1292--1331).

Through this correspondence they proved that mass-conserving solutions are unique on the full critical range \(0<m\le1\), but could establish their existence only for \(0<m<\tfrac12\).

We identify a one-sided, convexity-type invariant that holds for Bernstein-transform data and is propagated by their viscous scheme as a genuine maximum-principle bound.

We call it the half-slope invariant.

It sharpens the curvature barrier and thereby extends mass-conserving existence to the entire critical range \(0<m\le1\).

Hence \(m=1\) is the critical mass, confirming the threshold predicted by Vigil and Ziff (this http URL Interface Sci.133 (1989), no.1, 257--264).

The same invariant appears in the radial partial-mass formulation of the two-dimensional Keller--Segel equation, whose critical mass is \(8\pi\).