Density of Forces Producing Navier--Stokes Blowup

RESUMO

We study the density of smooth external forces for which the three-dimensional Navier--Stokes equations lose classical regularity by a prescribed time $T>0$. Starting from the compact forced blowup solution of OpenAI, we construct a blowup solution near every given smooth solution while preserving its initial velocity. A smooth cutoff of a local vector potential makes the given velocity vanish near the support of a rescaled blowup solution. The two velocities then have no nonlinear interaction, and the resulting force remains smooth through the blowup time. For fixed viscosity and zero initial velocity, such forces are dense in the relative $L^1_tH^s_x$ topology on both $\mathbb{T}^3$ and $\mathbb{R}^3$ exactly when $s<1/2$.