I believe a core element of the smoothness problem solution is that it does blow up in finite time, even with viscosity.
Perhaps an expert could provide a more accurate response, but here is how ChatGPT tried to explain it to me: "In this construction, the viscous term becomes large along with the acceleration, pressure-gradient, and nonlinear momentum-transfer terms, but they balance/cancel in a very precise way. That allows the velocity in the concentrating vortex core to grow without bound even though ν is nonzero."
Perhaps an expert could provide a more accurate response, but here is how ChatGPT tried to explain it to me: "In this construction, the viscous term becomes large along with the acceleration, pressure-gradient, and nonlinear momentum-transfer terms, but they balance/cancel in a very precise way. That allows the velocity in the concentrating vortex core to grow without bound even though ν is nonzero."