Establish global-in-time well-posedness for the wave-borne point vortex problem

Establish global-in-time well-posedness for the capillary-gravity wave-borne point vortex problem, including control of the Rayleigh–Taylor sign condition and prevention of collisions between the point vortex and the free surface.

Background

The paper formulates the wave-borne point vortex equations as a free-boundary Euler system coupled to the Helmholtz–Kirchhoff ordinary differential equation for the vortex center. Local well-posedness is available under a Rayleigh–Taylor sign condition, while previous work has shown that the condition can fail in finite time when the vortex approaches the free surface.

Because global existence is unresolved, the orbital stability theorem proved in the paper is explicitly conditional: it applies only on intervals where a solution exists and satisfies an appropriate a priori bound. A global well-posedness theory would remove this qualification and clarify the long-time dynamics of the constructed waves.