环面上的投影连续数据同化：共振和无核区域
Projectional continuous data assimilation on the torus: Resonant and kernel-free regimes
AI总结：
研究二维不可压缩纳维-斯托克斯方程在周期环面上的连续数据同化，通过单个带符号标量速度投影，利用移动框架展开等两种互补机制，在\(L^2\)和\(H^1\)中实现指数同步，扩展到特定观测，提升同步结果。
AI中文摘要：
我们研究了二维不可压缩纳维-斯托克斯方程在周期环面上的连续数据同化，使用在规定空间变化方向上的单个带符号标量速度投影。通过两种互补机制，我们证明了在\(L^2\)和\(H^1\)中的指数同步。在规则共振区域，通过移动框架展开、适配坐标和Farhat-Lunasin-Titi对数估计来控制非平凡不可见电流，前提是产生的几何剪切缺陷是粘性可吸收的。这恢复了恒定有理方向的周期单分量机制，并适用于真正的非恒定投影场。在无核区域，定性单射性和紧致性产生具有任意小粘性泄漏的可观测性，在没有增益上限限制的情况下，对于每个足够大的增益都能实现同步。对于足够规则的投影场，在通常的增益分辨率条件下，两种机制都扩展到满足一阶近似性质的\(L^2\)稳定I型粗标量观测。一个常见的抛物平滑论证将所有四个\(L^2\)同步结果提升到\(H^1\)同步，而无需额外的观测假设。
英文摘要：
We study continuous data assimilation for the two-dimensional incompressible Navier-Stokes equations on the periodic torus using a single signed scalar velocity projection in a prescribed spatially varying direction. We prove exponential synchronization in both L2 and H1 through two complementary mechanisms. In the regular resonant regime, nontrivial invisible currents are controlled through a moving-frame expansion, adapted coordinates, and Farhat-Lunasin-Titi logarithmic estimates, provided the resulting geometric shear defect is viscously absorbable. This recovers the periodic one-component mechanism for constant rational directions and applies to genuinely nonconstant projection fields. In the kernel-free regime, qualitative injectivity and compactness yield observability with arbitrarily small viscous leakage, giving synchronization for every sufficiently large gain without an upper gain restriction. For sufficiently regular projection fields, both mechanisms extend to L2-stable Type-I coarse scalar observations satisfying a first-order approximation property, under the usual gain-resolution condition. A common parabolic smoothing argument then upgrades all four L2-synchronization results to H1-synchronization without additional observation hypotheses.
