On the Global Regularity of a Helical-decimated Version of the 3D Navier-Stokes Equations

TL;DR

This paper proves global regularity for a helicity-decimated 3D Navier-Stokes model, showing solutions remain smooth for all time due to conserved quantities and new a priori estimates.

Contribution

It introduces a decimated Navier-Stokes model based on helicity that guarantees global regularity and uniqueness, supported by novel estimates linking helicity to solution smoothness.

Findings

Global existence and uniqueness of solutions in the decimated model.

Finite bounds on $H^{1/2}$ and $H^{3/2}$ norms over time.

Helicity conservation provides physical insight into regularity.

Abstract

We study the global regularity, for all time and all initial data in $H^{1/2}$, of a recently introduced decimated version of the incompressible 3D Navier-Stokes (dNS) equations. The model is based on a projection of the dynamical evolution of Navier-Stokes (NS) equations into the subspace where helicity (the $L^2-$scalar product of velocity and vorticity) is sign-definite. The presence of a second (beside energy) sign-definite inviscid conserved quadratic quantity, which is equivalent to the $H^{1/2}-$Sobolev norm, allows us to demonstrate global existence and uniqueness, of space-periodic solutions, together with continuity with respect to the initial conditions, for this decimated 3D model. This is achieved thanks to the establishment of two new estimates, for this 3D model, which show that the $H^{1/2}$ and the time average of the square of the $H^{3/2}$ norms of the velocity field remain finite. Such two additional bounds are known, in the spirit of the work of H. Fujita and T. Kato \cite{kato1,kato2}, to be sufficient for showing well-posedness for the 3D NS equations. Furthermore, they are directly linked to the helicity evolution for the dNS model, and therefore with a clear physical meaning and consequences.