Let ν>0 be a fixed viscosity constant. A vector field u0:R3→R3 satisfies the decaying initial velocity condition if it is smooth (C∞), divergence-free (∇⋅u0=0), and all derivatives of u0 decay faster than any polynomial at infinity: for every derivative order m∈N and every decay rate K∈R, there exists a constant C∈R such that ∥∂mu0(x)∥≤C/(1+∥x∥)K for all x∈R3. A function f:R3×[0,∞)→R3 satisfies the decaying force condition if it is smooth on R3×[0,∞) and all derivatives of f decay faster than any polynomial in both space and time: for every m∈N and K∈R, there exists C∈R such that ∥∂x,tmf(x,t)∥≤C/(1+∥x∥+t)K for all x∈R3 and t≥0. A pair of functions (v,p) with v:R3×[0,∞)→R3 and p:R3×[0,∞)→R is a global smooth solution with finite energy for initial data u0 and force f if: (i) v and p are smooth on R3×[0,∞); (ii) v(⋅,t) is square-integrable for each t≥0 and the kinetic energy ∫R3∥v(x,t)∥2dx remains uniformly bounded for all t≥0; (iii) the incompressibility condition ∇⋅v(⋅,t)=0 holds for all t≥0; (iv) the initial condition v(x,0)=u0(x) holds for all x∈R3; and (v) the Navier–Stokes equation ∂tv+(v⋅∇)v=νΔv−∇p+f holds for all x∈R3 and t≥0. Does there exist an initial velocity field u0 satisfying the decaying initial velocity condition and a force f satisfying the decaying force condition such that no pair (v,p) is a global smooth solution with finite energy for this data?