Uniqueness of the homogeneous cooling state for inelastic hard spheres
For a constant coefficient of normal restitution $\alpha \in (0, 1)$, define the three-dimensional inelastic hard-sphere collision operator weakly by $$\int Q_\alpha(F, F)\varphi\, dv = \frac{1}{2} \iiint_{\mathbb{R}^3 \times \mathbb{R}^3 \times S^2} |(v - w) \cdot \omega|\, F(v) F(w) \bigl[\varphi(v') + \varphi(w') - \varphi(v) - \varphi(w)\bigr]\, d\omega\, dv\, dw,$$ where $v' = v - \frac{1+\alpha}{2}((v - w) \cdot \omega)\,\omega$, $w' = w + \frac{1+\alpha}{2}((v - w) \cdot \omega)\,\omega$ and $d\omega$ is surface measure. Self-similar solutions of the spatially homogeneous inelastic Boltzmann equation $\partial_t f = Q_\alpha(f, f)$ (homogeneous cooling states) correspond to profiles solving $Q_\alpha(G, G) = \tau \nabla_v \cdot (vG)$ for some $\tau > 0$; for fixed mass, changing $\tau$ only rescales the velocity, so one may take $\tau = 1$. Mischler and Mouhot proved that for every $\alpha \in (0, 1)$ there is a smooth positive profile with given mass and vanishing momentum (2006), and that it is unique for $\alpha$ in an interval $[\alpha_*, 1)$ with a constructive $\alpha_* \in (0, 1)$ (2009), where they also proved convergence to it for initial data bounded in $L^1_3$, when $1 - \alpha$ is small enough depending on the mass, energy and that bound; this answers for small inelasticity a conjecture of Ernst and Brito on the attractivity of self-similar solutions.
Problem. For every $\alpha \in (0, 1)$, is there exactly one nonnegative density $F$ satisfying $$Q_\alpha(F, F) = \nabla_v \cdot (vF), \qquad \int F\, dv = 1, \qquad \int vF\, dv = 0$$ in the sense of distributions, with $\int (1 + |v|^3) F\, dv < \infty$ and $\int F |\log F|\, dv < \infty$? No rotational symmetry is imposed.
Source: Mischler and Mouhot (2006), §5, Conjecture 1. Stated by AIM — Open Applied Problems for three-dimensional hard spheres, in a specified class of profiles.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, granular gases, inelastic Boltzmann equations, self-similar solutions
References
@article{mischler2006,
author = {Mischler, St{\'e}phane and Mouhot, Cl{\'e}ment},
title = {Cooling process for inelastic {B}oltzmann equations for hard spheres. {P}art {II}: {S}elf-similar solutions and tail behavior},
journal = {J. Stat. Phys.},
volume = {124},
number = {2--4},
year = {2006},
pages = {703--746},
doi = {10.1007/s10955-006-9097-8},
eprint = {math/0607536},
archivePrefix = {arXiv},
}
@article{mischler2009,
author = {Mischler, St{\'e}phane and Mouhot, Cl{\'e}ment},
title = {Stability, convergence to self-similarity and elastic limit for the {B}oltzmann equation for inelastic hard spheres},
journal = {Comm. Math. Phys.},
volume = {288},
number = {2},
year = {2009},
pages = {431--502},
doi = {10.1007/s00220-009-0773-9},
eprint = {math/0701449},
archivePrefix = {arXiv},
}
@article{ernst2002,
author = {Ernst, M. H. and Brito, R.},
title = {Scaling solutions of inelastic {B}oltzmann equations with over-populated high energy tails},
journal = {J. Stat. Phys.},
volume = {109},
number = {3--4},
year = {2002},
pages = {407--432},
doi = {10.1023/A:1020437925931},
}
@misc{aim-inelastic-cooling-profile-uniqueness,
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 homogeneous cooling state for inelastic hard spheres}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/239-inelastic-cooling-profile-uniqueness.md},
note = {Entry 239 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-00601, "Uniqueness of the homogeneous cooling state for inelastic hard spheres", clopen.solutions.