AI 中文总结
本文针对一类非物理MHD离散化的两域边缘上同调系统，提出Hodge强制性判据，证明其全局动力学性质，并通过实例验证相关谱与能量结论。
AI 中文摘要
本文引入了一种针对无散边缘上同调的有限维两域系统，其MHD型命名仅指二次交换模式与精确总能量抵消，并非物理意义上的MHD离散化。研究将一般抵消类与修正后的显式实现区分开来：对于对角D(a)与反对称J，反对易子D(a)J+JD(a)为反对称，其投影双线性映射具备所需的三线性反对称性。核心结果为Hodge强制性判据：当且仅当其调和1-上同调空间为平凡时，完整无散空间才存在耗散性所需的Poincaré型估计；在此条件下，全局存在性、精确能量恒等式、吸收球及紧全局吸引子均成立。当存在调和模式时，调和解耦相互作用类会产生不变调和仿射纤维与逐纤维吸引子。确定性圆盘、环形及双孔示例验证了该谱判据、能量定律，以及一般调和交换与调和纤维不变性的区别。
英文摘要
A finite-dimensional two-field system for divergence-free edge cochains is introduced. Its MHD-type designation refers only to a quadratic exchange pattern and exact total-energy cancellation; it is not a physical MHD discretization. A general cancellation class is separated from a corrected explicit realization: the anticommutator $D(a)J+JD(a)$ is skew-symmetric for diagonal $D(a)$ and skew-symmetric $J$, and its projected bilinear map has the required trilinear antisymmetry. The central result is a Hodge coercivity criterion: the full divergence-free space admits the Poincaré-type estimate needed for dissipativity if and only if its harmonic $1$-cochain space is trivial. Under this condition, global existence, an exact energy identity, an absorbing ball, and a compact global attractor follow. When harmonic modes are present, a harmonic-decoupled interaction class yields invariant harmonic affine fibres and fibre-wise attractors. Deterministic disk, annular, and two-hole examples illustrate the spectral criterion, energy law, and distinction between general harmonic exchange and harmonic-fibre invariance.
