Conditional regularity of solutions of the three-dimensional Navier-Stokes equations and implications for intermittency

摘要

Two unusual time-integral conditional regularity results are presented for the three-dimensional Navier-Stokes equations. The ideas are based on L~(2m)-norms of the vorticity, denoted by ω_m(t), and particularly on D_m=[ω_o ~(-1)ω_m(t)]~(αm) where α_m = 2m/(4m - 3) for m ≥ 1. The first result, more appropriate for the unforced case, can be stated simply: if there exists an 1 m < ∞ for which the integral condition is satisfied (Z_m = D_m + _1/D_m): ∞_o,~t in (1+z_m/c_(4,m))then no singularity can occur on [0, t]. The constant c_(4, m) ↘ 2 for large m. Second, for the forced case, by imposing a critical lower bound on ∞+0,~t, no singularity can occur in D_m(t) for large initial data. Movement across this critical lower bound shows how solutions can behave intermittently, in analogy with a relaxation oscillator. Potential singularities that drive ∞_0,~t D_md τ over this critical value can be ruled out whereas other types cannot.