Smooth stationary water waves with exponentially localized vorticity

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. A great deal of recent activity has focused on finding waves with nontrivial vorticity ω. There are now many results on the existence of solutions to this problem for which the vorticity is non-vanishing at infinity, and several authors have constructed waves with ω having compact support. Our main theorem states that there are large families of stationary capillary-gravity waves that carry finite energy and exhibit an exponentially localized distribution of vorticity. They are solitary waves in the sense that the free surface is asymptotically flat. Remarkably, while their amplitude is small, the kinetic energy is O(1). In this and other respects, they are strikingly different from previously known rotational water waves.

To construct these solutions, we exploit a previously unobserved connection between the steady water wave problem on the one hand and singularly perturbed elliptic PDE on the other. Indeed, our result expands the study of spike-layer solutions to free boundary problems with physical relevance. ω. There are now many results on the existence of solutions to this problem for which the vorticity is non-vanishing at infinity, and several authors have constructed waves with ω having compact support. Our main theorem states that there are large families of stationary capillary-gravity waves that carry finite energy and exhibit an exponentially localized distribution of vorticity. They are solitary waves in the sense that the free surface is asymptotically flat. Remarkably, while their amplitude is small, the kinetic energy is O(1). In this and other respects, they are strikingly different from previously known rotational water waves.

To construct these solutions, we exploit a previously unobserved connection between the steady water wave problem on the one hand and singularly perturbed elliptic PDE on the other. Indeed, our result expands the study of spike-layer solutions to free boundary problems with physical relevance.