In this article, we pursue our investigation of the connections between the
theory of computation and hydrodynamics. We prove the existence of stationary
solutions of the Euler equations in Euclidean space, of Beltrami type, that can
simulate a universal Turing machine. In particular, these solutions possess
undecidable trajectories. Heretofore, the known Turing complete constructions
of steady Euler flows in dimension 3 or higher were not associated to a
prescribed metric. Our solutions do not have finite energy, and their
construction makes crucial use of the non-compactness of $\mathbb R^3$, however
they can be employed to show that an arbitrary tape-bounded Turing machine can
be robustly simulated by a Beltrami flow on $\mathbb T^3$ (with the standard
flat metric). This shows that there exist steady solutions to the Euler
equations on the flat torus exhibiting dynamical phenomena of (robust)
computational complexity as high as desired. We also quantify the energetic
cost for a Beltrami field on $\mathbb T^3$ to simulate a tape-bounded Turing
machine, thus providing additional support for the space-bounded Church-Turing
thesis. Another implication of our construction is that a Gaussian random
Beltrami field on Euclidean space exhibits arbitrarily high computational
complexity with probability $1$. Finally, our proof also yields Turing complete
flows and diffeomorphisms on $\mathbb{S}^2$ with zero topological entropy, thus
disclosing a certain degree of independence within different hierarchies of
complexity.
