Uniqueness of Glimm–Lax solutions for arbitrarily rough bounded data
Let $d \in \mathbb{R}^2$, $r > 0$ and $f \in C^4(B_r(d); \mathbb{R}^2)$, with $Df$ strictly hyperbolic and both characteristic fields genuinely nonlinear, and consider $$u_t + f(u)_x = 0, \qquad u(0) = u_0,$$ for measurable $u_0 : \mathbb{R} \to B_r(d)$, with $a = \|u_0 - d\|_\infty$. Glimm and Lax proved that when $a$ is small enough there is a global weak entropy solution, whose total variation on bounded intervals is finite at every $t > 0$ although that of $u_0$ may be infinite. The Glimm–Lax class consists of the bounded distributional solutions that attain $u_0$ in $L^1_{\mathrm{loc}}$ as $t \downarrow 0$, satisfy $\partial_t \eta(u) + \partial_x q(u) \le 0$ for every convex entropy $\eta$ with flux $q$, $Dq = D\eta\, Df$, and satisfy for all $t, L > 0$ $$\|u(t) - d\|_\infty \le C_0 \sqrt{a}, \qquad \sup_{b \in \mathbb{R}} \operatorname{TV}_{[b, b+L]} u(t) \le C_0 \Bigl(\sqrt{a} + \frac{L}{t}\Bigr),$$ where $C_0$ is the fixed Glimm–Lax constant associated with $f$ and $d$. Cheng, Faile and Krupa proved that for every $s > 0$ and $p \in [1, \infty]$ there is $\epsilon_1 > 0$ such that the class contains at most one solution with initial data $u_0$ whenever $u_0 \in W^{s,p}_{\mathrm{loc}}(\mathbb{R})$ and $a \le \epsilon_1$. Bressan, Marconi and Vaidya had proved well-posedness for a class of data whose infinite total variation is concentrated on finitely many open intervals of small total length.
Problem. Is there $\delta > 0$ such that, for every measurable $u_0 : \mathbb{R} \to B_r(d)$ with $a = \|u_0 - d\|_\infty \le \delta$, the Glimm–Lax class contains at most one solution with initial data $u_0$? No positive fractional Sobolev regularity of $u_0$ is assumed.
Source: Cheng, Faile and Krupa (2026), Open Problem #1, after the uniqueness question for the solutions of Glimm and Lax (1970) recorded by Bressan (2024), §8. Stated by AIM — Open Applied Problems for all small bounded measurable data.
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{cheng2026,
author = {Cheng, Jeffrey and Faile, Cooper and Krupa, Sam G.},
title = {The unique limit of the {G}limm--{L}ax construction for {S}obolev data and obstructions to 1-d convex integration},
journal = {Arch. Ration. Mech. Anal.},
volume = {250},
number = {5},
year = {2026},
pages = {Paper No. 82},
doi = {10.1007/s00205-026-02232-1},
eprint = {2601.01349},
archivePrefix = {arXiv},
}
@book{glimm1970,
author = {Glimm, James and Lax, Peter D.},
title = {Decay of solutions of systems of nonlinear hyperbolic conservation laws},
series = {Memoirs of the American Mathematical Society},
number = {101},
publisher = {American Mathematical Society},
address = {Providence, RI},
year = {1970},
doi = {10.1090/memo/0101},
}
@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{bressan2025,
author = {Bressan, Alberto and Marconi, Elio and Vaidya, Ganesh},
title = {Uniqueness domains for {$\mathbf{L}^\infty$} solutions of $2 \times 2$ hyperbolic conservation laws},
journal = {Arch. Ration. Mech. Anal.},
volume = {249},
number = {6},
year = {2025},
pages = {Paper No. 70},
doi = {10.1007/s00205-025-02147-3},
eprint = {2505.00420},
archivePrefix = {arXiv},
}
@misc{aim-glimm-lax-measurable-data-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 Glimm–Lax solutions for arbitrarily rough bounded data}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/360-glimm-lax-measurable-data-uniqueness.md},
note = {Entry 360 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-00597, "Uniqueness of Glimm–Lax solutions for arbitrarily rough bounded data", clopen.solutions.