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

Abstract

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.

3 figures and 3 tables

References in corpus (4)

• Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations
• Solution properties of a 3D stochastic Euler fluid equation
• Phase-field model for the Rayleigh--Taylor instability of immiscible fluids
• A regularity criterion for solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations and associated computations

Cited by in corpus (2)

• How close are shell models to the 3D Navier-Stokes equations?
• An analytical and computational study of the incompressible Toner-Tu Equations