The limit of vanishing viscosity for the incompressible 3D Navier-Stokes equations with helical symmetry

Abstract

In this paper, we are concerned with the vanishing viscosity problem for the three-dimensional Navier-Stokes equations with helical symmetry, in the whole space. We choose viscosity-dependent initial $\bu_0^\nu$ with helical swirl, an analogue of the swirl component of axisymmetric flow, of magnitude $\mathcal{O}(\nu)$ in the $L^2$ norm; we assume $\bu_0^\nu \to \bu_0$ in $H^1$. The new ingredient in our analysis is a decomposition of helical vector fields, through which we obtain the required estimates.