Finite time blowup for an averaged three-dimensional Navier-Stokes equation

The paper constructs an averaged three-dimensional Navier-Stokes equation whose bilinear term retains the usual cancellation law and energy identity but admits finite-time blow-up solutions. This demonstrates that any abstract method relying solely on upper-bound function-space estimates and the energy identity cannot prove global regularity for the true Navier-Stokes system.