The Fefferman-Stein decomposition for the Constantin-Lax-Majda equation: Regularity criteria for inviscid fluid dynamics revisited

The celebrated Beale-Kato-Majda (BKM) criterion for the 3D Euler equations has been updated by Kozono and Taniuchi by replacing the supremum with the bounded mean oscillation norm. We consider this generalized criterion in an attempt to understand it more intuitively by giving an alternative explanation. For simplicity, we first treat the Constantin-Lax-Majda (CLM) equation∂ω∂t=H(ω)ω for the vorticity ω in one-dimension and identify a mechanism underlying the update of such an estimate. We consider a Fefferman-Stein (FS) decomposition for the initial vorticity ω = ω0 + H[ω1] and how it propagates under the dynamics of the CLM equation. In particular, we obtain a set of dynamical equations for it, which reads in its simplest case ∂ω0∂t=ω0H[ω0]−ω1H[ω1] and ∂ω1∂t=ω0H[ω1]+ω1H[ω0]. The equation for the second component ω1, responsible for a possible logarithmic blow-up, is linear and homogeneous; hence it remains zero if it is so initially until a stronger blow-up takes place. This rules out a logarithmic blow-up on its own and underlies the generalized BKM criterion. Numerical results are also presented to illustrate how each component of the FS decomposition evolves in time. Higher dimensional cases are also discussed. Without knowing fully explicit FS decompositions for the 3D Euler equations, we show that the second component of the FS decomposition will not appear if it is zero initially, thereby precluding a logarithmic blow-up.