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The Beale-Kato-Majda theorem contains a single criterion that controls thebehaviour of solutions of the $3D$ incompressible Euler equations. Versions ofthis theorem are discussed in terms of the regularity issues surrounding the$3D$ incompressible Euler and Navier-Stokes equations together with aphase-field model for the statistical mechanics of binary mixtures called the$3D$ Cahn-Hilliard-Navier-Stokes (CHNS) equations. A theorem of BKM-type isestablished for the CHNS equations for the full parameter range. Moreover, forthis latter set, it is shown that there exists a Reynolds number and a bound onthe energy-dissipation rate that, remarkably, reproduces the $Re^{3/4}$ upperbound on the inverse Kolmogorov length normally associated with theNavier-Stokes equations alone. An alternative length-scale is introduced anddiscussed, together with a set of pseudo-spectral computations on a $128^{3}$grid.

The role of BKM-type theorems in $3D$ Euler, Navier-Stokes and Cahn-Hilliard-Navier-Stokes analysis