Steady Contiguous Vortex-Patch Dipole Solutions of the 2D Incompressible Euler Equation

De Huang; Jiajun Tong

Steady Contiguous Vortex-Patch Dipole Solutions of the 2D Incompressible Euler Equation

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arxiv 2406.09849 v3 pith:IFIUF2BG submitted 2024-06-14 math.AP physics.flu-dyn

De Huang , Jiajun Tong This is my paper · ORCID

classification math.APphysics.flu-dyn

keywords dipolevortex-patchboundarycontiguousequationeulerincompressiblepatch

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We rigorously construct the first steady traveling wave solutions of the 2D incompressible Euler equation that take the form of a contiguous vortex-patch dipole, which can be viewed as the vortex-patch counterpart of the well-known Lamb-Chaplygin dipole. Our construction is based on a novel fixed-point approach that determines the patch boundary as the fixed point of a certain nonlinear map. Smoothness and other properties of the patch boundary are also obtained.

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Stability for multiple Lamb dipoles
math.AP 2025-07 conditional novelty 7.0 of 10

Finite sums of Lamb dipoles in the half-plane, with ordered speeds and well-separated initial positions, are Lyapunov stable under the 2D Euler equations.

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