A two-phase fluid system — oil and water, say — evolves according to the Navier-Stokes-Cahn-Hilliard equations. At fine scales, the interface between phases wrinkles, pinches, breaks. At coarse scales, you see only a blurred average. A single coarse observation is consistent with many possible fine-scale states. The inverse problem — reconstructing the fine state from the coarse observation — is ill-posed.
Sun showed that continuous data assimilation resolves this. Instead of inverting a single observation, the method feeds a stream of coarse observations back into the model through a nudging term — a gentle persistent correction that pushes the model state toward consistency with whatever the observations show. Over time, the nudged model synchronizes with the true fine-scale evolution.
The mathematics is precise: the difference between the nudged solution and the true solution decays exponentially in time, provided the observation operator satisfies an H²-type approximation property and the nudging strength exceeds a computable threshold. The decay rate depends on observation resolution, but convergence is guaranteed even for very coarse observations — as long as they keep coming.
This is a statement about how information accumulates. A single blurry snapshot is ambiguous — many fine-scale states produce the same coarse appearance. But a sequence of blurry snapshots is not ambiguous, because different fine-scale states evolve differently. The dynamics break the degeneracy. Two states that look identical at time t will diverge by time t + dt, and the next coarse observation distinguishes them. The stream does what the snapshot cannot.
The through-claim: temporal continuity is a resolution enhancer. Blurriness in space is compensated by persistence in time. The nudging doesn't add spatial detail — it adds temporal constraint. And temporal constraint, applied continuously, is sufficient to pin down the spatial detail that no single observation can provide. Patience resolves what precision cannot.