Weak vorticity formulation of the incompressible 2D Euler equations in bounded domains

Abstract

In this article we examine the interaction of incompressible 2D flows with compact material boundaries. Our focus is the dynamic behavior of the circulation of velocity around boundary components and the possible exchange between flow vorticity and boundary circulation in flows with vortex sheet initial data. We formulate our results for flows outside a finite number of smooth obstacles. Our point of departure is the observation that ideal flows with vortex sheet regularity have well-defined circulations around connected components of the boundary. In addition, we show that the velocity can be uniquely reconstructed from the vorticity and boundary component circulations, which allows to recast 2D Euler evolution using vorticity and the circulations as dynamic variables. The weak form of this vortex dynamics formulation of the equations is called the weak vorticity formulation. Our first result is existence of a solution for the weak velocity formulation with vortex sheet initial data for flow outside a finite number of smooth obstacles. The proof is a straightforward adaptation of Delort's original existence result and requires the usual sign condition. The main result in this article is the equivalence between the weak velocity and weak vorticity formulations, without sign assumptions. Next, we focus on weak solutions obtained by mollifying initial data and passing to the limit, with the portion of vorticity singular with respect to the Lebesgue measure assumed to be nonnegative. For these solutions we prove that the circulations around each boundary component cannot be smaller than the initial data circulation, so that nonnegative vorticity may be absorbed by the boundary, but not produced by the boundary. In addition, we prove that if the weak solution conserves circulation at the boundary components it is a boundary coupled weak solution, a stronger version of the weak vorticity formulation. We prove existence of a weak solution which conserves circulation at the boundary components if the initial vorticity is integrable, i.e. if the singular part vanishes. In addition, we discuss the definition of the mechanical force which the flow exerts on material boundary components and its relation with conservation of circulation. Finally, we describe the corresponding results for a bounded domain with holes, and the adaptations required in the proofs.

Vorticity Measures and the Inviscid Limit

“…Now, χω ∞ ∈ BM∩H −1 (Ω), hence the result in [12,Proposition 4.8], see also [11,Proposition 2.10], applies to ω k I : ω k I is bounded in L 1 (Ω), ω k I ⇀ χω ∞ weak − * BM(Ω) and any weak− * limit in BM(Ω), µ, of |ω k I | is a continuous measure,…”

“…Now we give the proof of Lemma 1, which is a result on the equivalence between the weak velocity formulation and the interior weak vorticity formulation. The argument is based on the proofs of equivalence contained in [11] and [12], with variations due to the fact that the vorticity is only a bounded Radon measure locally, in the interior of the fluid domain. When regarded as a distribution in the entire fluid domain, the best regularity for ω ∞ is H −1 , which is the same as for wild solutions.…”

“…Because time is frozen at t we omit it hereafter. We adapt what was done in [12, Theorem 6.1 and Proposition 6.2], see also [11,Theorem 3.4 and Proposition 3.5], to the situation we have, where only interior estimates are available. Let ρ k be a cutoff away from the boundary.…”

Asymptotic behavior of 2D incompressible ideal flow around small disks

“…The purpose of this section is to introduce basic notation and recall facts about the Biot-Savart law, which expresses the velocity in terms of the vorticity and circulations. The details can be found, for example, in [15,22,14].…”

“…The above integral is considered in the counter-clockwise sense, henceτ = −n ⊥ , wheren denotes the outward unit normal vector at ∂Ω k . Vorticity together with the circulations uniquely determine the velocity, see, for example, [14] for a full discussion.…”