  In this article we construct a compact Riemannian manifold of high dimension
on which the time dependent Euler equations are Turing complete. More
precisely, the halting of any Turing machine with a given input is equivalent
to a certain global solution of the Euler equations entering a certain open set
in the space of divergence-free vector fields. In particular, this implies the
undecidability of whether a solution to the Euler equations with an initial
datum will reach a certain open set or not in the space of divergence-free
fields. This result goes one step further in Tao's programme to study the
blow-up problem for the Euler and Navier-Stokes equations using fluid
computers. As a remarkable spin-off, our method of proof allows us to give a
counterexample to a conjecture of Moore dating back to 1998 on the
non-existence of analytic maps on compact manifolds that are Turing complete.
