Vanishing nonlinear viscosity without commuting transport and diffusion matrices
Let $f \in C^4(U; \mathbb{R}^N)$, $N \ge 2$, on an open set $U$, and suppose $Df$ is strictly hyperbolic, with each characteristic field genuinely nonlinear or linearly degenerate. Suppose there is a strictly convex entropy $\eta$, with entropy flux $q$ satisfying $Dq = D\eta \, Df$. Let $B \in C^3(U; \mathbb{R}^{N \times N})$ be symmetric and uniformly positive definite, and assume $\operatorname{sym}(D^2 \eta \, B)$ is uniformly positive definite near a fixed state $u_*$.
For strictly hyperbolic systems with initial data of small total variation, Bianchini and Bressan proved that the solutions of the artificial-viscosity approximation $u_t + Df(u) u_x = \varepsilon u_{xx}$ are global, satisfy total variation bounds uniform in $\varepsilon$, and converge as $\varepsilon \to 0$ to the trajectories of a Lipschitz semigroup of weak solutions of $u_t + f(u)_x = 0$ satisfying the Liu admissibility conditions. Haspot and Jana extended the uniform total variation bounds to a state-dependent viscosity matrix $B(u)$ that commutes with $Df(u)$: for Temple-class systems, for $2 \times 2$ triangular systems, and, in a preprint, for general strictly hyperbolic systems, with convergence to the unique weak solution in the conservative case.
Problem. For all sufficiently small $\operatorname{TV}(u_0)$, with $u_0 - u_* \in L^1(\mathbb{R})$ and range near $u_*$, must the solutions of $$\partial_t u^\varepsilon + \partial_x f(u^\varepsilon) = \varepsilon \partial_x \bigl( B(u^\varepsilon) \partial_x u^\varepsilon \bigr), \qquad u^\varepsilon(0) = u_0,$$ satisfy $\sup_{t \ge 0} \operatorname{TV}(u^\varepsilon(t)) \le C \operatorname{TV}(u_0)$ uniformly for $0 < \varepsilon \le 1$, and converge in $C([0, T]; L^1)$ for every finite $T$ to the standard small-BV entropy semigroup solution? The matrices $B(u)$ and $Df(u)$ need not commute.
Source: Bressan (2024), §5, Open Problem #2, for positive semidefinite viscosity matrices. Stated by AIM — Open Applied Problems for symmetric, uniformly positive definite viscosity matrices compatible with a strictly convex entropy, without a commutation assumption.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, BV solutions, hyperbolic conservation laws, vanishing viscosity
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{bianchini2005,
author = {Bianchini, Stefano and Bressan, Alberto},
title = {Vanishing viscosity solutions of nonlinear hyperbolic systems},
journal = {Ann. of Math. (2)},
volume = {161},
number = {1},
year = {2005},
pages = {223--342},
doi = {10.4007/annals.2005.161.223},
}
@article{haspot2025,
author = {Haspot, Boris and Jana, Animesh},
title = {Vanishing viscosity limit for hyperbolic system of {T}emple class in 1-d with nonlinear viscosity},
journal = {Nonlinear Anal. Real World Appl.},
volume = {85},
year = {2025},
pages = {104346},
doi = {10.1016/j.nonrwa.2025.104346},
eprint = {2407.12766},
archivePrefix = {arXiv},
}
@article{haspot2026triangular,
author = {Haspot, Boris and Jana, Animesh},
title = {Viscous approximation of triangular system in 1-{D} with nonlinear viscosity},
journal = {Discrete Contin. Dyn. Syst.},
volume = {54},
year = {2026},
pages = {206--259},
doi = {10.3934/dcds.2026065},
eprint = {2503.04640},
archivePrefix = {arXiv},
}
@misc{haspot2026,
author = {Haspot, Boris and Jana, Animesh},
title = {Vanishing viscosity limit for $n \times n$ hyperbolic system of conservation laws in 1-d with nonlinear viscosity: {P}art-{I} {U}niform {BV} estimates},
year = {2025},
eprint = {2512.15620},
archivePrefix = {arXiv},
note = {Revised May 2026 (v2)},
}
@misc{aim-noncommuting-viscosity-limit,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Vanishing nonlinear viscosity without commuting transport and diffusion matrices}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/359-noncommuting-viscosity-limit.md},
note = {Entry 359 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-00622, "Vanishing nonlinear viscosity without commuting transport and diffusion matrices", clopen.solutions.