Study of the Navier–Stokes regularity problem with critical norms

We study the basic problems of regularity of the Navier–Stokes equations. The blowup criteria on the basis of the critical ${H}^{1/2}$-norm, is bounded from above by a logarithmic function, (Robinson et al 2012 J. Math. Phys. 53 115618). Assuming that the Cauchy–Schwarz inequality for the ${H}^{1/2}$-norm is not an overestimate, we replace it by a square-root of a product of the energy and the enstrophy. We carry out a simple asymptotic analysis to determine the time evolution of the energy. This generalises the (already ruled-out) self-similar blowup ansatz. Some numerical results are also presented, which support the above-mentioned replacement. We carry out a similar analysis for the four-dimensional Navier–Stokes equations.