Projectional continuous data assimilation on the torus: Resonant and kernel-free regimes

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.