Finite-time blowup for the Fourier-restricted Euler and hypodissipative Navier-Stokes model equations

TL;DR

This paper introduces Fourier-restricted Euler and hypodissipative Navier-Stokes equations, proving finite-time blowup for certain symmetric solutions in specific regimes, advancing understanding of singularity formation in fluid models.

Contribution

It defines new restricted Fourier mode models and proves finite-time blowup for symmetric solutions in inviscid and certain hypodissipative cases, with energy and enstrophy identities preserved.

Findings

Finite-time blowup proven for specific solutions.

Models respect energy and enstrophy identities.

Blowup occurs under symmetry and spectral constraints.

Abstract

In this paper, we introduce the Fourier-restricted Euler and hypodissipative Navier--Stokes equations. These equations are analogous to the Euler and hypodissipative Navier--Stokes equations respectively, but with the Helmholtz projection replaced by a projection onto a more restrictive constraint space; the $(u\cdot\nabla)u$ nonlinearity is otherwise unchanged. The constraint space restricts the divergence-free velocity to specific Fourier modes, which have a dyadic shell structure, and are constructed iteratively using permutations. In the inviscid case -- and in the hypo-viscous case when $\alpha<\frac{\log(3)}{6\log(2)} \approx .264$ -- we prove finite-time blowup for a set of solutions with a discrete group of symmetries. Our blowup Ansatz is odd, permutation symmetric, and mirror symmetric about the plane $x_1+x_2+x_3=0$. The Fourier-restricted Euler and hypodissipative Navier--Stokes equations respect both the energy equality and the identity for enstrophy growth from the full Euler and hypodissipative Navier--Stokes equations respectively, which is a substantial advance over the previous literature on Euler and Navier--Stokes model equations.