A note on helicity conservation in Leray models of incompressible flow (Q458368)

The authors consider the helical quantity (called regularized helicity) \(H_L(t):=\int_{\Omega}u\cdot w \, dx\), where \(w\) is the solution of the linear regularized vorticity equation, and \(u\) is the (given) strong solution of the Leray-\(\alpha\) model NEWLINE\[NEWLINE w_t + \bar{u}\cdot \nabla w - w\cdot \nabla u +\nabla \lambda - \nu \Delta w = \nabla \times f; \;\;\nabla \cdot w =0; \;\;w(0)=\nabla \times u_0. NEWLINE\]NEWLINE \(\Omega\) is assumed to be a periodic box \(\subset {\mathbb{R}}^3\). For the Navier-Stokes equations \(H_L = H\) (the usual helicity). The purpose of this paper is to show that the Leray-\(\alpha\) model conserves the \(H_L\) (while it does not conserve the \(H\)). In addition, this conservation extends to the discrete setting with a Crank-Nicolson time discretization and finite element spatial discretization.