# Smooth stationary water waves with exponentially localized vorticity

## TL;DR

This paper demonstrates the existence of large families of stationary water waves with finite energy and exponentially localized vorticity, using advanced mathematical techniques from elliptic equations and water wave theory.

## Contribution

It introduces a novel construction of stationary water waves with localized vorticity, expanding understanding of wave solutions with specific vorticity distributions.

## Key findings

- Existence of large families of such waves.
- Waves carry finite energy.
- Vorticity is exponentially localized.

## Abstract

We study stationary capillary-gravity waves in a two-dimensional body of water that rests above a flat ocean bed and below vacuum. This system is described by the Euler equations with a free surface. Our main result states that there exist large families of such waves that carry finite energy and exhibit an exponentially localized distribution of (nontrivial) vorticity. This is accomplished by combining ideas drawn from the theory of spike-layer solutions to singularly perturbed elliptic equations, with techniques from the study of steady solutions of the water wave problem.