Regimes of nonlinear depletion and regularity in the 3D Navier-Stokes equations

Abstract

The periodic $3D$ Navier-Stokes equations are analyzed in terms of dimensionless, scaled, $L^{2m}$-norms of vorticity $D_m$ ($1\le m<\infty$). The first in this hierarchy, $D_1$, is the global enstrophy. Three regimes naturally occur in the $D_1-D_m$ plane. 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 \cite{DGGKPV13}\,; 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.

9 figures and 3 tables

References in corpus (3)

• The multifractal nature of turbulent energy dissipation
• Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations
• On the strength of the nonlinearity in isotropic turbulence

Cited by in corpus (14)

• Self-attenuation of extreme events in Navier-Stokes turbulence
• Characterization of velocity-gradient dynamics in incompressible turbulence using local streamline geometry
• Revisiting turbulence small-scale behavior using velocity gradient triple decomposition
• Extreme Vortex States and the Growth of Enstrophy in 3D Incompressible Flows
• Weak and strong solutions of the $3D$ Navier-Stokes equations and their relation to a chessboard of convergent inverse length scales
• Maximum Amplification of Enstrophy in 3D Navier-Stokes Flows
• Systematic Search For Extreme and Singular Behavior in Some Fundamental Models of Fluid Mechanics
• Depletion of Nonlinearity in Magnetohydrodynamic Turbulence: Insights from Analysis and Simulations
• A regularity criterion for solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations and associated computations
• Nonlinear effects in buoyancy-driven variable density turbulence
• An analytical and computational study of the incompressible Toner-Tu Equations
• Phase transitions in the fractional three-dimensional Navier-Stokes equations
• Searching for Singularities in Navier-Stokes Flows Based on the Ladyzhenskaya-Prodi-Serrin Conditions
• Bounds on solutions of the rotating, stratified, incompressible, non-hydrostatic, three-dimensional Boussinesq equations