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Finite-time blowup for a Navier–Stokes model equation for the self-amplification of strain

Evan Miller

Vol. 16 (2023), No. 4, 997–1032
Abstract

We consider a model equation for the Navier–Stokes strain equation which has the same identity for enstrophy growth and a number of the same regularity criteria as the full Navier–Stokes strain equation, and is also an evolution equation on the same constraint space. We prove finite-time blowup for this model equation, which shows that the identity for enstrophy growth and the strain constraint space are not sufficient on their own to guarantee global regularity for Navier–Stokes. The mechanism for the finite-time blowup of this model equation is the self-amplification of strain, which is consistent with recent research suggesting that strain self-amplification, not vortex stretching, is the main mechanism behind the turbulent energy cascade. Because the strain self-amplification model equation is obtained by dropping certain terms from the full Navier–Stokes strain equation, we will also prove a conditional blowup result for the full Navier–Stokes equation involving a perturbative condition on the terms neglected in the model equation.

Keywords
Navier–Stokes, finite-time blowup
Mathematical Subject Classification
Primary: 35Q30
Milestones
Received: 31 January 2021
Revised: 21 July 2021
Accepted: 16 September 2021
Published: 15 June 2023
Authors
Evan Miller
Department of Mathematics
University of British Columbia
Vancouver
Canada

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