Velocity-Field Detection of Computation in Forced Navier–Stokes Flows
We construct smooth forces for three-dimensional incompressible Navier–Stokes flow, from rest at any fixed positive computable viscosity, whose velocity field detects whether a prescribed machine halts. On the unit flat torus, a pointwise test of the third velocity component uses successively shorter stirring intervals in a fixed spatial region. On $\mathbb R^2\times\mathbb T$, an integral test uses an expanding array of translation regions and a force with globally bounded mixed derivatives. The vertical velocity solves an advection–diffusion equation: short bursts control its pointwise…
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