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

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 $(u\cdot \mathrm{\nabla })u$ 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.