Uniqueness of the Brownian coagulation similarity profile
For cluster masses $x, y > 0$ let $$K(x,y) = 2 + (x/y)^{1/3} + (y/x)^{1/3} = \bigl(x^{1/3} + y^{1/3}\bigr)\bigl(x^{-1/3} + y^{-1/3}\bigr),$$ Smoluchowski's kernel for Brownian coagulation. A profile $F$ gives a self-similar solution $f(t,x) = t^{-2} F(x/t)$ of Smoluchowski's coagulation equation with this kernel when, for every $\varphi \in C_c^\infty(0,\infty)$, $$\int_0^\infty F(x)\bigl[x\varphi'(x) - \varphi(x)\bigr]\,dx = \frac12 \int_0^\infty\!\!\int_0^\infty K(x,y) F(x) F(y) \bigl[\varphi(x+y) - \varphi(x) - \varphi(y)\bigr]\,dx\,dy.$$ For the constant kernel $K \equiv 2$ the profile of mass one is $e^{-x}$. Existence of self-similar profiles for non-solvable kernels such as this one was proved by Escobedo, Mischler and Rodríguez Ricard and by Fournier and Laurençot. The kernel has the form $2 + \varepsilon W$ with $W(x,y) = (x/y)^{1/3} + (y/x)^{1/3}$ and $\varepsilon = 1$. For homogeneous kernels $2 + \varepsilon W$ with $0 \le W(x,y) \le (x/y)^\alpha + (y/x)^\alpha$, $\alpha < 1/2$, and further analyticity assumptions on $W$, Niethammer, Throm and Velázquez proved that profiles of given mass are unique when $\varepsilon$ is small enough; Throm proved the same for continuous $W$ with $\alpha < 1$ (assuming a matching lower bound on $W$ when $\alpha \ge 1/2$). For bounded $W$ and small $\varepsilon$, Cañizo and Throm proved that solutions approach a unique self-similar profile.
Problem. Is there exactly one nonnegative profile $F$ with $\int_0^\infty x F(x)\,dx = 1$ and $\int_0^\infty \bigl(1 + x + x^{-1/3}\bigr) F(x)\,dx < \infty$ satisfying the equation above for every $\varphi \in C_c^\infty(0,\infty)$, profiles being equal when they agree almost everywhere?
Source: Stated by AIM — Open Applied Problems for Smoluchowski's Brownian kernel, after the uniqueness question for self-similar profiles of non-solvable kernels; see Niethammer, Throm and Velázquez (2015) and Throm (2021).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, coagulation equations, self-similar solutions, Smoluchowski's equation
References
@misc{niethammer2015,
author = {Niethammer, Barbara and Throm, Sebastian and Vel{\'a}zquez, Juan J. L.},
title = {A revised proof of uniqueness of self-similar profiles to {S}moluchowski's coagulation equation for kernels close to constant},
year = {2015},
eprint = {1510.03361},
archivePrefix = {arXiv},
note = {Revised 2017 (v3); announced in J. Stat. Phys. 164 (2016), 399--409, doi:10.1007/s10955-016-1553-5},
}
@article{throm2021,
author = {Throm, Sebastian},
title = {Stability and uniqueness of self-similar profiles in {$L^1$} spaces for perturbations of the constant kernel in {S}moluchowski's coagulation equation},
journal = {Comm. Math. Phys.},
volume = {383},
number = {3},
year = {2021},
pages = {1361--1407},
doi = {10.1007/s00220-021-03967-6},
eprint = {1902.10000},
archivePrefix = {arXiv},
}
@article{escobedo2005,
author = {Escobedo, Miguel and Mischler, St{\'e}phane and Rodr{\'\i}guez Ricard, Mariano},
title = {On self-similarity and stationary problem for fragmentation and coagulation models},
journal = {Ann. Inst. H. Poincar{\'e} Anal. Non Lin{\'e}aire},
volume = {22},
number = {1},
year = {2005},
pages = {99--125},
doi = {10.1016/j.anihpc.2004.06.001},
}
@article{fournier2005,
author = {Fournier, Nicolas and Lauren{\c{c}}ot, Philippe},
title = {Existence of self-similar solutions to {S}moluchowski's coagulation equation},
journal = {Comm. Math. Phys.},
volume = {256},
number = {3},
year = {2005},
pages = {589--609},
doi = {10.1007/s00220-004-1258-5},
}
@article{canizo2021,
author = {Ca{\~n}izo, Jos{\'e} A. and Throm, Sebastian},
title = {The scaling hypothesis for {S}moluchowski's coagulation equation with bounded perturbations of the constant kernel},
journal = {J. Differential Equations},
volume = {270},
year = {2021},
pages = {285--342},
doi = {10.1016/j.jde.2020.07.036},
eprint = {1910.08015},
archivePrefix = {arXiv},
}
@misc{aim-brownian-coagulation-profile,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniqueness of the Brownian coagulation similarity profile}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/163-brownian-coagulation-profile.md},
note = {Entry 163 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00849, "Uniqueness of the Brownian coagulation similarity profile", clopen.solutions.