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

Highlights: • A BKM-type theorem on the regularity of solutions of the $3D$ CHNS equations is discussed. • A length scale associated with the energy dissipation rate is also discussed. • Numerical solutions on a 128 cubed grid are displayed. The Beale–Kato–Majda theorem contains a single criterion that controls the behaviour of solutions of the $3D$ incompressible Euler equations. Versions of this theorem are discussed in terms of the regularity issues surrounding the $3D$ incompressible Euler and Navier–Stokes equations together with a phase-field model for the statistical mechanics of binary mixtures called the $3D$ Cahn–Hilliard–Navier–Stokes (CHNS) equations. A theorem of BKM-type is established for the CHNS equations for the full parameter range. Moreover, for this latter set, it is shown that there exists a Reynolds number and a bound on the energy-dissipation rate that, remarkably, reproduces the $R{e}^{3∕4}$ upper bound on the inverse Kolmogorov length normally associated with the Navier–Stokes equations alone. An alternative length-scale is introduced and discussed, together with a set of pseudo-spectral computations on a $128^3$ grid.