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

We study the critical coagulation–fragmentation equation with multiplicative coagulation kernel $a(s,\widehat{s})=s\widehat{s}$ and constant fragmentation kernel $b(s,\widehat{s})=1$. Under the Bernstein transform, mass-conserving solutions correspond to solutions of a singular Hamilton–Jacobi equation studied by Tran and Van (Comm. Pure Appl. Math. 75 (2022), no. 6, 1292–1331). Through this correspondence they proved that mass-conserving solutions are unique on the full critical range $0<m\le 1$, but could establish their existence only for $0<m<\frac{1}{2}$. 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 $0<m\le 1$. Hence $m=1$ is the critical mass, confirming the threshold predicted by Vigil and Ziff (J. Colloid 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 $8\pi$.