Structure of ENO Entropy Dissipation: Parity Dichotomy for the ENO--TV Conjecture and Shift Cohomology
Abstract
Essentially non-oscillatory (ENO) reconstruction provides a key mechanism for designing high-order entropy-stable schemes for hyperbolic conservation laws, with its sign property ensuring nonnegative local dissipation for a prescribed entropy. However, two fundamental questions concerning convergence remain open: whether this dissipation provides the coercivity required for weak-BV compactness, as posited by the ENO--TV conjecture, and whether entropy stability transfers from the prescribed entropy pair to additional pairs. This paper resolves the ENO--TV conjecture by establishing a sharp parity dichotomy: it holds if and only if the reconstruction order k=2 or k is odd, and fails for all even orders k≥ 4.The key to our proof is a localization principle that eliminates dependence on nonlinear adaptive stencil selection, establishing a two-sided equivalence between ENO dissipation and a canonical finite-difference functional. For odd orders, the conjecture is proved via a hidden quadratic energy and novel discrete Gagliardo--Nirenberg inequalities. For even orders k≥ 4, ENO null modes, on which ENO dissipation vanishes, yield counterexamples that disprove the conjecture. This dichotomy extends to quasi-uniform meshes, but for every k≥ 2, the conjecture can fail on non-quasi-uniform meshes. Addressing the above second open question, we discover on ENO null modes that local entropy transfer is governed by the first cohomology of a unipotent shift. Using apolar duality and binary covariants, we compute the dimensions of the associated cohomology subspaces and prove that smooth local entropy transfer encounters generic obstructions for every k≥ 4. By revealing how ENO null modes link global coercivity and local entropy compatibility, this work provides a structural foundation for the compactness and convergence analysis of high-order entropy-stable discretizations.
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BibTeX
@misc{li2026structure,
title = {{Structure of ENO Entropy Dissipation: Parity Dichotomy for the ENO--TV Conjecture and Shift Cohomology}},
author = {Zhuoyun Li and Kailiang Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.19283},
eprint = {2607.19283},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.19283},
}
RIS
TY - UNPB
TI - Structure of ENO Entropy Dissipation: Parity Dichotomy for the ENO--TV Conjecture and Shift Cohomology
AU - Zhuoyun Li
AU - Kailiang Wu
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2607.19283
UR - https://arxiv.org/abs/2607.19283
AB - Essentially non-oscillatory (ENO) reconstruction provides a key mechanism for designing high-order entropy-stable schemes for hyperbolic conservation laws, with its sign property ensuring nonnegative local dissipation for a prescribed entropy. However, two fundamental questions concerning convergence remain open: whether this dissipation provides the coercivity required for weak-BV compactness, as posited by the ENO--TV conjecture, and whether entropy stability transfers from the prescribed entropy pair to additional pairs. This paper resolves the ENO--TV conjecture by establishing a sharp parity dichotomy: it holds if and only if the reconstruction order k=2 or k is odd, and fails for all even orders k≥ 4.The key to our proof is a localization principle that eliminates dependence on nonlinear adaptive stencil selection, establishing a two-sided equivalence between ENO dissipation and a canonical finite-difference functional. For odd orders, the conjecture is proved via a hidden quadratic energy and novel discrete Gagliardo--Nirenberg inequalities. For even orders k≥ 4, ENO null modes, on which ENO dissipation vanishes, yield counterexamples that disprove the conjecture. This dichotomy extends to quasi-uniform meshes, but for every k≥ 2, the conjecture can fail on non-quasi-uniform meshes. Addressing the above second open question, we discover on ENO null modes that local entropy transfer is governed by the first cohomology of a unipotent shift. Using apolar duality and binary covariants, we compute the dimensions of the associated cohomology subspaces and prove that smooth local entropy transfer encounters generic obstructions for every k≥ 4. By revealing how ENO null modes link global coercivity and local entropy compatibility, this work provides a structural foundation for the compactness and convergence analysis of high-order entropy-stable discretizations.
ER -