Gain of regularity for water waves with surface tension

TL;DR

This paper demonstrates that surface tension induces a regularizing effect on water waves, causing the curvature of the fluid surface to instantaneously gain smoothness under certain initial conditions.

Contribution

It establishes a rigorous mathematical result showing instantaneous smoothing of water waves with surface tension, using energy estimates and invariant vector fields.

Findings

Surface tension causes instant regularization of water wave surfaces.

Initial curvature in Sobolev space leads to higher smoothness over time.

The proof employs energy estimates and invariant vector fields.

Abstract

Regularizing effects of surface tension are studied for interfacial waves between a two-dimensional, infinitely-deep and irrotational flow of water and vacuum. The water wave problem under the influence of surface tension is formulated as a system of second-order in time nonlinear dispersive equations. The main result states that if the curvature of the initial fluid surface is in the Sobolev space of order k+7/2 and if its derivatives decay faster than a polynomial of degree k+1 does then the curvature of the fluid surface corresponding to the solution of the problem instantaneously gains k/2 derivatives of smoothness compared to the initial state. The proof uses energy estimates under invariant vector fields of the associated linear operator.