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

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 Omega(m)(t), and particularly on D-m=[pi(-1)(0)Omega(m)(t)](alpha m), where alpha(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 < infinity for which the integral condition is satisfied (Z(m) = D-m (+) (1)/D-m): integral(t)(0) ln (1+Z(m)/c(4,m)) d tau >= 0 , then no singularity can occur on [0,t]. The constant c(4, m) SE arrow 2 for large m. Second, for the forced case, by imposing a critical lower bound on integral(t)(0) D-m d tau, 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 integral(t)(0) D-m d tau over this critical value can be ruled out whereas other types cannot. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4742857]