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 $k\ge 4$.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 $k\ge 4$, ENO null modes, on which ENO dissipation vanishes, yield counterexamples that disprove the conjecture. This dichotomy extends to quasi-uniform meshes, but for every $k\ge 2$, 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 $k\ge 4$. 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.

79 pages. Expanded with stronger results on entropy transfer. The resolution of ENO-TV conjecture remains unchanged