Global Existence and Uniqueness of Weak Solutions of Three-Dimensional Euler Equations with Helical Symmetry in the Absence of Vorticity Stretching

Abstract

We prove uniqueness and existence of the weak solutions of Euler equations with helical symmetry, with initial vorticity in L ∞ under "no vorticity stretching" geometric constraint. Our article follows the argument of the seminal work of Yudovich. We adjust the argument to resolve the difficulties which are specific to the helical symmetry.

Planar Limits of Three-Dimensional Incompressible Flows with Helical Symmetry

“…As shown in [6] for instance, a (smooth) function p = p(r, θ, z) is a helical function if and only if, when expressed in the (r, ξ, η) variables, it is independent of ξ: p = q(r, σ 2π θ + z), for some q = q(r, η) Similarly, a (smooth) vector field u is helical if and only if there exist v r , v θ , v z , functions of (r, η) such that u r = v r (r,…”

“…In this section we discuss symmetry reduction and the limit σ → ∞ for the Euler equations under an additional geometric assumption, considered already in [5,6]. This assumption can be viewed as the analog of the no-swirl condition in axisymmetric flows and for this reason we will call it the no helical swirl or no helical stretching condition.…”

“…We now briefly review these results, referring the reader to [5,6] for more details. We will then discuss the limit problem as σ → ∞ and converge of solutions.…”

“…As a matter of fact, an additional geometric condition is imposed on inviscid flows, akin to assuming no swirl in the axisymmetric setting, which we call no helical swirl or no helical stretching. Under this condition, B. Ettinger and E. Titi [6] showed global existence and uniqueness of weak solutions in an appropriate vorticity-stream function formulation. This formulation can be used, because, even for finite σ, the flow is essentially twodimensional, in the sense that it is completely determined by the dynamics of the first two components of the velocity field restricted to any cross section of the pipe.…”

“…(For a discussion about uniqueness of weak solutions, , within the class of all LerayHopf weak solutions of the three-dimensional Navier-Stokes with helical initial data, see [1].) The situation is different, and rather interesting, in the case of ideal fluid governed by the Euler equations, see [5,6]. As a matter of fact, an additional geometric condition is imposed on inviscid flows, akin to assuming no swirl in the axisymmetric setting, which we call no helical swirl or no helical stretching.…”