Remarks on stationary and uniformly rotating vortex sheets: flexibility results

Abstract

In this paper, we construct new, uniformly rotating solutions of the vortex sheet equation bifurcating from circles with constant vorticity amplitude. The proof is accomplished via a Lyapunov–Schmidt reduction and a second-order expansion of the reduced system. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.

Traveling Waves Near Couette Flow for the 2D Euler Equation

“…In particular, following the approach of Burbea [11], there has been many works concerning the existence of single or multiple patches and their degenerate case of point vortices not only for 2D Euler equation but also for other two-dimensional active scalar equations such as the generalized surface quasi-geostrophic equation through contour dynamics equations paired with bifurcation theory, desingularization techniques and variational tools. We refer to the interested reader, for more details, to the non-exhaustive list [12,13,14,15,26,28,29,39,41,47,49,50,55,56,60,59] and the references therein.…”

Existence of stationary vortex sheets for the 2D incompressible Euler equation

“…For all j ≥ 1 and j ∈ N * , there hold− ∫ cos( jα) sin(θ − α) 4 sin 2 ( θ−α 2 ) jα) sin(θ − α) 4 sin 2 ( θ−α 2 ) jθ) − cos( jα) 4 sinProof Identities (4.4) and (4.6) were proved in Lemma A.8[24]. Indeed, (4.4) can be deduced from the identity− ∫ cos( jα) sin(θ − α) 4 sin ( jθ))(θ),where H(⋅) is the Hilbert transform on torus and hence H(cos( jθ)) = sin( jθ).…”

“…Identities (4.4) and (4.6) were proved in Lemma A.8 [24]. Indeed, (4.4) can be deduced from the identity where is the Hilbert transform on torus and hence .…”

“…A generalization of the rotating segment is the Protas–Sakajo class [40], which is made out of segments rotating about a common center of rotation with endpoints at the vertices of a regular polygon. Recently, a new class of vortex sheet was obtained in [24] via degenerate bifurcation from rotating circles. Note that the existence of nontrivial steady vortex sheet in is not apparent in view of the rigidity results obtained in [23], where the authors showed for uniformly rotating vortex sheets with angular velocity and strength , only trivial solutions exist.…”

Editorial: Mathematical problems in physical fluid dynamics: part II

“…Gomez-Serrano et al . [ 1 ] find new relative equilibria involving vortex sheets that bifurcate from circular vortex sheets with constant magnitude. Ceci & Seis [ 2 ] study the evolution of two-dimensional Euler flows with highly concentrated vorticity and establish new connections to point-vortex dynamics.…”