Uniqueness of entropy-admissible renormalized Boltzmann solutions
On $\mathbb{T}^3_x \times \mathbb{R}^3_v$ consider the Boltzmann equation $\partial_t f + v \cdot \nabla_x f = Q(f, f)$ with hard-sphere collisions, $$Q(f, f)(v) = \int_{\mathbb{R}^3} \int_{\mathbb{S}^2} |(v - v_*) \cdot \omega|\, \bigl[f(v') f(v_*') - f(v) f(v_*)\bigr]\, d\omega\, dv_*,$$ where $v' = v - [(v - v_*) \cdot \omega]\,\omega$ and $v_*' = v_* + [(v - v_*) \cdot \omega]\,\omega$, all factors taken at the same $(t, x)$. A renormalized solution in the DiPerna–Lions class of finite mass, energy and entropy is a nonnegative $f$ such that, on every finite time interval, $\int f\,(1 + |v|^2 + |\log f|)\, dx\, dv$ is bounded, $Q(f, f)/(1 + f) \in L^1_{\mathrm{loc}}$, and $$(\partial_t + v \cdot \nabla_x)\, \beta(f) = \beta'(f)\, Q(f, f)$$ holds in the sense of distributions for every $C^1$ function $\beta$ with $|\beta'(z)| \le C/(1 + z)$. Here it is also required to have an initial trace in $L^1$, to conserve mass and momentum, and to satisfy the energy inequality and the entropy inequality $$H(f_t) + \int_0^t D(f_s)\, ds \le H(f_0), \qquad H(f) = \int f \log f,$$ where $D(f) = -\int Q(f, f) \log f \ge 0$ is the entropy dissipation, taken as its nonnegative collision integral. DiPerna and Lions proved global existence and weak stability of renormalized solutions on $\mathbb{R}^3_x \times \mathbb{R}^3_v$ for initial data of finite mass, energy and entropy and finite second moment in $x$.
Problem. Are two nonnegative renormalized solutions in this class with the same initial datum necessarily equal almost everywhere?
Source: A question about the renormalized solutions of DiPerna and Lions (1989); stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Boltzmann equation, renormalized solutions, uniqueness
References
@article{diperna1989,
author = {DiPerna, R. J. and Lions, P.-L.},
title = {On the {C}auchy problem for {B}oltzmann equations: global existence and weak stability},
journal = {Ann. of Math. (2)},
volume = {130},
number = {2},
year = {1989},
pages = {321--366},
doi = {10.2307/1971423},
}
@misc{aim-renormalized-boltzmann-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 entropy-admissible renormalized Boltzmann solutions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/227-renormalized-boltzmann-uniqueness.md},
note = {Entry 227 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove nonuniqueness for the three-dimensional periodic hard-sphere Boltzmann equation among global entropy solutions satisfying exact local conservation of mass, momentum, and kinetic energy. We construct one nonnegative initial density with bounded velocity support and finite mass, energy, and absolute entropy that gives rise to two distinct such solutions. Both are strongly continuous in \(L^1\), and their collision gain and …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00710, "Uniqueness of entropy-admissible renormalized Boltzmann solutions", clopen.solutions.