Universal computation in forced Navier–Stokes flows

We realize finite reciprocal affine instruction maps by smooth incompressible shear flows on a flat three-torus. Applied to a reversible one-head recorder with a finite transition table, this gives a complete machine-to-fluid construction: a fixed particle enters a fixed open strip exactly when a given machine halts, with zero initial velocity, zero pressure, and a solenoidal mean-zero force that is periodic after initialization. The construction acts on full closed rectangles and controls every intermediate trajectory. Separate heights resolve overlap between sources and targets, and an endpoint-size estimate controls all excursions independently of the reciprocal scaling factor.

We construct smooth external forces for incompressible Navier–Stokes flow on a fixed flat three-torus at any fixed positive computable viscosity, from zero initial velocity, such that a fixed particle enters a fixed open set exactly when a prescribed Turing machine halts. Every mixed derivative of the force and velocity is bounded and square-integrable in time in spatial supremum norm. The construction uses the machine's ordinary instructions, records their history in a third coordinate, and compensates for finer spatial gates by longer time steps.

At every fixed positive computable viscosity, we construct smooth mean-zero forces on the flat three-torus that become stationary after time one and make a fixed particle, initially in a fluid at rest, enter a fixed open set exactly when a prescribed Turing machine halts. The force has bounded derivatives of every order and a finite effective description. A reversible recording table is realized on whole planar rectangles by Hamiltonian motions, then driven by a mean-zero spatial clock. Separate choices give periodic forcing from time zero or a force whose derivatives are square-integrable in time.

We construct smooth solenoidal mean-zero forces, periodic from time zero, for which a fixed particle on a flat three-torus reaches a fixed open strip exactly when a given machine halts. The initial fluid velocity and the pressure are zero. We also realize positive diagonal maps on coding sheets by incompressible shears, with explicit normal compensation for changes of planar area, and reciprocal maps on whole boxes of positive thickness. Complete local inverses, initialization rules, and intermediate trajectory bounds connect these geometric constructions to finite computations.

Finite prefix instructions may change area and erase information. We give explicit history processors and smooth incompressible motions that retain that information and realize every instruction on its full domain. A first application assigns each machine and finite input a smooth mean-zero force on the flat unit three-torus, at any fixed positive computable viscosity. The solution starts from rest; a fixed particle enters a fixed open strip exactly when the machine halts. The force repeats with period one after an initial loading interval. We then prove alternative realizations using full boxes, normal compensation, invariant planes, and a spatial clock. The constructions specify their initialization, comparison class, derivative bounds, and continuous-time observation. Exact formulas and effective cutoffs provide finite descriptions of the fields and all their derivatives.

At any fixed positive computable viscosity, smooth forces can make a fixed fluid particle detect the halting of an arbitrary machine, starting from rest, while every mixed derivative of the force and velocity decreases faster than every inverse power of time. We give three complete memory constructions: compact moving curls, alternating fractional coordinates, and a periodic lattice on the flat three-torus. Each machine step takes one unit of physical time. The constructions retain earlier records at separated spatial scales and use an exact open detector with a shrinking signal.

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 error in the first construction, and a moving cutoff controls the total escaped mass in the second.

We construct smooth forces of fixed compact spatial support for three-dimensional incompressible Navier–Stokes flow from rest whose particle at the origin detects halting by entering a fixed half-space. Every mixed derivative of the force and velocity decays at rate $O((1+t{)}^{-1-j})$ for time order j. Three scalar potentials lift a coded rectangle, perform its instruction, and lower its image; an unbounded logarithmic clock supplies the decay. We also realize area-changing prefix instructions on an invariant torus plane, and construct a planar Hamiltonian processor admitting periodic forcing, stationary forcing after startup, and a velocity-field detector through a companion diffusion theorem.

We realize finite positive diagonal affine maps of determinant one by effective smooth incompressible flows on neighborhoods of entire closed rational solid boxes. The source and target families are each disjoint, but may overlap each other. The construction uses localized curls, evacuation to storage and obstacle detours. A balanced three-stack recorder then assigns every machine and finite input a smooth Navier–Stokes force with one compact spatial support, periodic after a loading interval, at any fixed positive computable viscosity. The fluid starts at rest, and one fixed particle enters one fixed open cube exactly when the machine halts. Further constructions give fixed torus charts, periodicity from time zero, alternative history guards and bounded or slab observers. Onto slow clocks yield separate decaying forces. Each construction includes an all-time observation proof, effective derivative bounds and a stated pressure comparison class.