Can one-dimensional entropy-dissipating systems have two bounded solutions?
Consider a system of conservation laws $w_t + f(w)_x = 0$ in one space dimension, with states in an open convex set $U \subset \mathbb{R}^N$, a smooth strictly hyperbolic flux $f$, and a smooth entropy pair $(\eta, q)$ with $D^2\eta > 0$ and $Dq = D\eta\, Df$. For strictly hyperbolic systems whose characteristic fields are each genuinely nonlinear or linearly degenerate, Bressan and Guerra proved that, in the presence of a strictly convex entropy, every entropy weak solution taking values in the domain of the standard semigroup of small-total-variation solutions coincides with a trajectory of that semigroup. Bressan records a $3 \times 3$ strictly hyperbolic system, with bounded measurable initial data, that has infinitely many solutions but admits no convex entropy. Krupa constructed, for a $p$-system, Riemann data with infinitely many bounded solutions, each vacuously satisfying the Lax and Liu E-conditions; his method provably does not admit a strictly convex entropy.
Problem. Do there exist an integer $N \ge 2$, an open convex state set $U \subset \mathbb{R}^N$, a smooth flux $f : U \to \mathbb{R}^N$ with $Df(u)$ having $N$ distinct real eigenvalues for every $u \in U$, and a smooth entropy pair $(\eta, q)$ satisfying $$D^2\eta(u) > 0, \qquad Dq(u) = D\eta(u)\, Df(u),$$ together with $T > 0$ and bounded measurable initial data $u_0$, for which two distinct bounded distributional solutions $u, v : (0, T) \times \mathbb{R} \to K$, for a compact $K \subset U$, satisfy $$w_t + f(w)_x = 0, \qquad \partial_t \eta(w) + \partial_x q(w) \le 0 \qquad (w = u, v),$$ and both attain $u_0$ strongly in $L^1_{\mathrm{loc}}$ at $t = 0$? Distinct means that they differ on a set of positive spacetime measure. The entropy inequality must hold in distributions throughout spacetime, not only at jump discontinuities; no bounded-variation hypothesis is imposed.
Source: Bressan (2024), Section 8, Open Problem 6. Stated by AIM — Open Applied Problems for bounded solutions with values in a compact subset of a convex state set, a strong initial trace, and the entropy inequality in distributions.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, entropy solutions, hyperbolic conservation laws
References
@article{bressan2024,
author = {Bressan, Alberto},
title = {One-dimensional hyperbolic conservation laws: past and future},
journal = {J. Hyperbolic Differ. Equ.},
volume = {21},
number = {3},
year = {2024},
pages = {523--561},
doi = {10.1142/S0219891624400022},
eprint = {2310.16707},
archivePrefix = {arXiv},
}
@article{bressanguerra2024,
author = {Bressan, Alberto and Guerra, Graziano},
title = {Unique solutions to hyperbolic conservation laws with a strictly convex entropy},
journal = {J. Differential Equations},
volume = {387},
year = {2024},
pages = {432--447},
doi = {10.1016/j.jde.2024.01.005},
eprint = {2305.10737},
archivePrefix = {arXiv},
}
@article{krupa2026,
author = {Krupa, Sam G.},
title = {Non-uniqueness and {BV} blowup for vacuously {L}iu-admissible solutions to $p$-system via computer-assisted proof},
journal = {J. Math. Fluid Mech.},
volume = {28},
number = {2},
year = {2026},
pages = {37},
doi = {10.1007/s00021-026-01018-5},
eprint = {2403.07784},
archivePrefix = {arXiv},
}
@misc{aim-one-dimensional-convex-entropy-nonuniqueness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Can one-dimensional entropy-dissipating systems have two bounded solutions?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/361-one-dimensional-convex-entropy-nonuniqueness.md},
note = {Entry 361 at commit 24edaad, 2026-10-15},
}
Resolution claims
In this paper, we consider $2\times2$ hyperbolic systems of conservation laws in one spatial dimension. We use the computational ansatz introduced in the hyperbolic theory by the author and Székelyhidi to study the constitutive set corresponding to the PDE. The rank-one convex geometry of this set relates to non-uniqueness and the existence of low-regularity solutions. Through a computer-assisted search, we …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: arXiv:2609.09353v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00815, "Can one-dimensional entropy-dissipating systems have two bounded solutions?", clopen.solutions.