Steady vortex patch with polygonal symmetry for the planar Euler equations in a disc. (February 2020)

• Record Type: Journal Article
• Authors: Cao, Daomin; Wan, Jie; Wang, Guodong
• Is Part Of: Nonlinear analysis. Volume 51(2020)
• Publication Date: 2020-02
• Subjects: Fluid dynamics -- Incompressible Euler flows -- Vortex patch; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2019.103008
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: In this paper, we construct two types of vortex patch equilibria for the two-dimensional Euler equations in a disc. The first type is called the " N + 1 type" equilibrium, in which a central vortex patch is surrounded by N identical patches with opposite signs, and the other type is called the " 2 N type" equilibrium, in which the centers of N identical positive patches and N negative patches lie evenly on a circle. The construction is performed by solving a variational problem for the vorticity in which the kinetic energy is maximized subject to some symmetry constraints, and then analyzing the asymptotic behavior as the vorticity strength goes to infinity.