发表机构
Southern University of Science and Technology（南方科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
解决ENO-TV猜想这一离散强制问题，证明其奇偶二分法，奇数k≥3估计成立偶数k≥4不成立。通过局部化等方法，还给出相关强制估计，计算上同调空间维数，揭示不同阶在不同网格上的强制性情况。
AI 中文摘要
我们解决了ENO-TV猜想，这是双曲守恒律熵稳定逼近的紧致性理论中的一个离散强制问题。对于从紧支单元平均值进行的k阶本质无振荡（ENO）重构，该猜想询问非负ENO源乘以振幅的(k - 1)次幂是否能均匀控制(k + 1)次绝对跳跃矩。我们证明了一个奇偶二分法：对于奇数k≥3估计成立，对于偶数k≥4不成立；已知的二阶情况完成了分类。局部化给出了一个与源均匀可比的与选择无关的有限差分泛函，并将猜想简化为离散插值。对于奇数阶，分部求和揭示了一个隐藏的平方；一个离散的加利亚尔多 - 尼伦伯格不等式产生强制性。对于偶数阶，泛函多项式核中的欧拉多项式块产生反例，这些反例在任意小的扰动下仍然存在，使得所有受影响的ENO比较严格。我们还证明了对于每个k≥2的两个强制估计：控制大于振幅固定分数的跳跃以及局部块模至多k - 2次采样多项式。通过凯莱 - 西尔维斯特分解，我们计算了多项式跳跃轮廓上晶格移位的齐次一阶上同调空间的维数。在四阶时，对于三次通量和全局严格凸熵，简化的熵 - 通量失配的总七次分量在至多二次轮廓上表示一个非零类，因此在零常数状态下没有平移不变的有限模板C^7局部原函数。奇数阶强制性在全局拟均匀网格上持续存在，而对于每个k≥2，它在固定的不规则网格上不成立，尽管每个界面贡献仍然是非负的。这种失败是由于网格几何形状。
英文摘要
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.
Comments79 pages. Expanded with stronger results on entropy transfer. The resolution of ENO-TV conjecture remains unchanged
