The role of sign and the incompressible Navier–Stokes equation

Consider the incompressible axially symmetric Navier–Stokes equations. Let z-axis be the symmetric axis. As a possible blowup scenario, the fluid flows may move towards the z-axis and the radial velocity component may satisfy a negative sign condition, as time approaches a possible blowup time T. In this paper, we construct solutions to the axially symmetric Navier–Stokes equations whose radial velocity has a negative sign. Those solutions have infinite energy, but are globally smooth. Surprisingly, the “dangerous” sign condition plays an essential positive role in proving the global regularity of such solutions. Our result may have some implications on potential blowup scenarios of incompressible Navier–Stokes equations when a sign condition holds in a local dangerous area.