Continuity of finite-cluster mass at the onset of Flory gelation
Let $0 < \alpha < 1$ and let $K : (0,\infty)^2 \to [0,\infty)$ be continuous and symmetric. Assume that for some $0 < c < C < \infty$ and some continuous $\ell > 0$, $$c(x^\alpha y + x y^\alpha) \le K(x,y) \le C(x^\alpha y + x y^\alpha), \qquad \lim_{y \to \infty} \frac{K(x,y)}{y} = \ell(x), \qquad c x^\alpha \le \ell(x) \le C x^\alpha.$$ Let $\mu_0$ be a finite nonnegative measure on $(0,\infty)$ with $\int x \, d\mu_0 = 1$. A Flory solution is a family $(\mu_t)_{t \ge 0}$ of finite nonnegative measures on $(0,\infty)$ whose particle number $\mu_t((0,\infty))$ and mass $M_1(t) = \int x \, d\mu_t$ are nonincreasing, such that for every $\phi \in C_c([0,\infty))$ and every $t \ge 0$ $$\langle \mu_t, \phi \rangle = \langle \mu_0, \phi \rangle + \frac12 \int_0^t \!\! \iint K(x,y) \bigl[\phi(x+y) - \phi(x) - \phi(y)\bigr] \, d\mu_s(x) \, d\mu_s(y) \, ds - \int_0^t \bigl(1 - M_1(s)\bigr) \langle \mu_s, \ell \phi \rangle \, ds.$$ Its gelation time is $T_g = \inf\{t \ge 0 : M_1(t) < 1\}$. Fournier and Laurençot showed that every Flory solution has $T_g \le \langle \mu_0, x^{1-\alpha} \rangle / \bigl((1 - 2^{-\alpha}) c\bigr)$, and that $M_1$ is continuous and strictly decreasing on $(T_g, \infty)$, with $M_1(t) \to 0$ as $t \to \infty$.
Problem. For every such solution whose gelation time satisfies $0 < T_g < \infty$, must $M_1$ be continuous at $T_g$, with $M_1(T_g) = \lim_{t \downarrow T_g} M_1(t) = 1$?
Source: Fournier and Laurençot (2009), the question after Proposition 2.3, for 0 < α ≤ 1; stated for 0 < α < 1 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, coagulation equations, gelation
References
@article{fournier2009,
author = {Fournier, Nicolas and Lauren{\c{c}}ot, Philippe},
title = {{M}arcus--{L}ushnikov processes, {S}moluchowski's and {F}lory's models},
journal = {Stochastic Process. Appl.},
volume = {119},
number = {1},
year = {2009},
pages = {167--189},
doi = {10.1016/j.spa.2008.02.003},
eprint = {0706.2057},
archivePrefix = {arXiv},
}
@misc{aim-flory-gelation-mass-continuity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Continuity of finite-cluster mass at the onset of Flory gelation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/383-flory-gelation-mass-continuity.md},
note = {Entry 383 at commit 24edaad, 2026-10-15},
}
Resolution claims
We prove continuity of the finite-cluster mass at the gelation time for every weak Flory solution with a continuous symmetric kernel comparable to $x^\alpha y+xy^\alpha$, where $0<\alpha<1$, and a finite initial measure of unit first moment. No higher initial moment is required. A hypothetical jump forces a positive amount of mass into a finite band at sizes of order $(T_g-t)^{-1/\alpha}$. …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/flory_onset.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00793, "Continuity of finite-cluster mass at the onset of Flory gelation", clopen.solutions.