Regimes of nonlinear depletion and regularity in the 3D Navier–Stokes equations Academic Article uri icon

abstract

  • © 2014 IOP Publishing Ltd & London Mathematical Society. The periodic 3D Navier-Stokes equations are analyzed in terms of dimensionless, scaled, L2m-norms of vorticity Dm(1≥m<∞). The first in this hierarchy, D1, is the global enstrophy. Three regimes naturally occur in the D1-Dmplane. Solutions in the first regime, which lie between two concave curves, are shown to be regular, owing to strong nonlinear depletion. Moreover, numerical experiments have suggested, so far, that all dynamics lie in this heavily depleted regime [1]; new numerical evidence for this is presented. Estimates for the dimension of a global attractor and a corresponding inertial range are given for this regime. However, two more regimes can theoretically exist. In the second, which lies between the upper concave curve and a line, the depletion is insufficient to regularize solutions, so no more than Leray's weak solutions exist. In the third, which lies above this line, solutions are regular, but correspond to extreme initial conditions. The paper ends with a discussion on the possibility of transition between these regimes.

altmetric score

  • 0.5

author list (cited authors)

  • Gibbon, J. D., Donzis, D. A., Gupta, A., Kerr, R. M., Pandit, R., & Vincenzi, D.

citation count

  • 14

publication date

  • September 2014