On Symmetric Water Waves with Constant Vorticity

Abstract

We prove that a solution to the gravity water wave problem with constant vorticity, whose wave profile as well as its horizontal velocity component at the free surface are symmetric at any instant of time, is given by a traveling wave. The proof is based on maximum principles and structural properties of the governing equations.

Keywords:

• Travelling waves
• constant vorticity
• symmetry

2000 Mathematics Subject Classification:

• 35Q31
• 42B37
• 76B15