A universal quantitative error bound for small-BV hyperbolic flows
Let $f \in C^4(U; \mathbb{R}^N)$ be strictly hyperbolic near $0 \in U$, with every characteristic field genuinely nonlinear, and let $(\eta, q)$ be a uniformly strictly convex entropy pair, $Dq = D\eta\, Df$. Write $S_t u_0$ for the standard small-total-variation entropy solution of $u_t + f(u)_x = 0$ on $\mathbb{R}$. Fix $T, R, M > 0$ and a sufficiently small $\delta > 0$. For $0 < \varepsilon < e^{-1}$ consider every approximate flow $u : [0, T] \to L^1(\mathbb{R}; U)$ supported in $[-R, R]$ with $\sup_t \operatorname{TV} u(t) \le \delta$, $\|u(0) - u_0\|_1 \le \varepsilon$ and $\|u(t) - u(s)\|_1 \le M|t - s| + \varepsilon$. For a scalar test function $\phi \in C_c^1(\mathbb{R}^2)$ let $$\mathcal{R}_{g,h}(s, t; \phi) = \int g(u(s,x))\phi(s,x)\,dx - \int g(u(t,x))\phi(t,x)\,dx + \int_s^t \!\! \int [g(u)\phi_t + h(u)\phi_x]\,dx\,dr,$$ and assume, for all $0 \le s < t \le T$, $$|\mathcal{R}_{\mathrm{id},f}| \le \varepsilon(t - s + \varepsilon)\|\phi\|_{W^{1,\infty}}, \qquad \mathcal{R}_{\eta,q} \ge -\varepsilon(t - s + \varepsilon)\|\phi\|_{W^{1,\infty}} \quad (\phi \ge 0).$$ Bressan and Guerra proved that, for such systems, every entropy weak solution with values in the domain of the semigroup is a semigroup trajectory; with compactness this gives a nondecreasing modulus $\varrho$ with $\varrho(0) = 0$ such that every such approximation satisfies $\|u(t) - S_t u_0\|_1 \le \varrho(\varepsilon)$ on $[0, T]$. For vanishing viscosity approximations with viscosity $\varepsilon$, Bressan and Yang proved an $L^1$ error of order $(1 + t)\operatorname{TV}(u_0)\sqrt{\varepsilon}\,|\ln\varepsilon|$.
Problem. Is there $C = C(f, \eta, U, \delta, T, R, M)$ such that every such approximation satisfies $$\sup_{0 \le t \le T} \|u(t) - S_t u_0\|_1 \le C\sqrt{\varepsilon}\,|\log\varepsilon|?$$
Source: Bressan (2024), Section 7, Open Problem 3, which asks for the asymptotics of the universal convergence rate and suggests ε^{1/2}|ln ε| in the genuinely nonlinear case. Stated by AIM — Open Applied Problems as an upper bound of that order.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- NA
- Keywords
- AIM Open Applied Problems, a posteriori error estimates, 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{bressanyang2004,
author = {Bressan, Alberto and Yang, Tong},
title = {On the convergence rate of vanishing viscosity approximations},
journal = {Comm. Pure Appl. Math.},
volume = {57},
number = {8},
year = {2004},
pages = {1075--1109},
doi = {10.1002/cpa.20030},
}
@misc{aim-universal-hyperbolic-error-rate,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A universal quantitative error bound for small-BV hyperbolic flows}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/367-universal-hyperbolic-error-rate.md},
note = {Entry 367 at commit 24edaad, 2026-10-15},
}
Resolution claims
We give a negative answer to a proposed universal error estimate for approximate small-BV hyperbolic flows. For a smooth strictly hyperbolic two-component system with both fields genuinely nonlinear and a uniformly strictly convex entropy, we construct one fixed initial datum and approximations satisfying the prescribed conservation and entropy residual bounds on every time strip. Their $L^1$ error divided by $\sqrt\varepsilon\,|\log\varepsilon|$ …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-20 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/paper.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00841, "A universal quantitative error bound for small-BV hyperbolic flows", clopen.solutions.