Existence of corotating and counter-rotating vortex pairs for active scalar equations

Abstract

In this paper, we study the existence of corotating and counter-rotating pairs of simply connected patches for Euler equations and the $(SQG{)}_{\alpha }$ equations with $\alpha \in (0,1).$ From the numerical experiments implemented for Euler equations in \cite{DZ, humbert, S-Z} it is conjectured the existence of a curve of steady vortex pairs passing through the point vortex pairs. There are some analytical proofs based on variational principle \cite{keady, Tur}, however they do not give enough information about the pairs such as the uniqueness or the topological structure of each single vortex. We intend in this paper to give direct proofs confirming the numerical experiments and extend these results for the $(SQG{)}_{\alpha }$ equation when $\alpha \in (0,1)$. The proofs rely on the contour dynamics equations combined with a desingularization of the point vortex pairs and the application of the implicit function theorem.

39 pages, we unified some sections

References in corpus (3)

• On the V-states for the generalized quasi-geostrophic equations
• The centrally symmetric V-states for active scalar equations. Two-dimensional Euler with cut-off
• Doubly connected V-states for the planar Euler equations

Cited by in corpus (23)

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• Global bifurcation of rotating vortex patches
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• Kármán Vortex Street in incompressible fluid models
• Symmetry in stationary and uniformly-rotating solutions of active scalar equations
• Global bifurcation for corotating and counter-rotating vortex pairs
• Non uniform rotating vortices and periodic orbits for the two-dimensional Euler Equations
• Two-Front Solutions of the SQG Equation and its Generalizations
• Vortex patches choreography for active scalar equations
• Existence and regularity of co-rotating and travelling global solutions for the generalized SQG equation
• Steady Contiguous Vortex-Patch Dipole Solutions of the 2D Incompressible Euler Equation
• Existence of co-rotating and travelling vortex patches with doubly connected components for active scalar equations
• Steady vortex patches near a nontrivial irrotational flow
• Existence of corotating asymmetric vortex pairs for Euler equations
• Existence and stability of smooth traveling circular pairs for the generalized surface quasi-geostrophic equation
• On Singular Vortex Patches, II: Long-time dynamics
• Dynamic Behavior of a Multi-Layer Quasi-Geostrophic Model: Weak and Time-Periodic Solutions
• Slow traveling-wave solutions for the generalized surface quasi-geostrophic equation
• Existence of stationary vortex sheets for the 2D Euler equation
• Rotating vortex patches for the planar Euler equations in a disk
• Time quasi-periodic vortex patches for quasi-geostrophic shallow-water equations
• Existence and Uniqueness for the SQG Vortex-Wave System when the Vorticity is Constant near the Point-Vortex
• Co-rotating and traveling vortex sheets for the 2D incompressible Euler equation