Nonlinear Water Waves on Uniform Current in Lagrangian Coordinates

On Nonlinear Water Waves in a Channel

on nonlinear water waves in a channel

A Lagrangian for water waves

A Lagrangian for strongly nonlinear unsteady water waves ~including overturning waves! is obtained. It is shown that the system of quadratic equations for the Stokes coefficients, which determine the shape of a steady wave ~discovered by Longuet-Higgins 100 years after Stokes derived his system of cubic equations! directly follows from the canonical system of Lagrange equations. Applications to...

Two Dimensional Water Waves in Holomorphic Coordinates

Abstract. This article is concerned with the infinite depth water wave equation in two space dimensions. We consider this problem expressed in position-velocity potential holomorphic coordinates. Viewing this problem as a quasilinear dispersive equation, we establish two results: (i) local well-posedness in Sobolev spaces, and (ii) almost global solutions for small localized data. Neither of th...

Lagrangian Averaging, Nonlinear Waves, and Shock Regularization

In this thesis, we explore various models for the flow of a compressible fluid as well as model equations for shock formation, one of the main features of compressible fluid flows. We begin by reviewing the variational structure of compressible fluid mechanics. We derive the barotropic compressible Euler equations from a variational principle in the material frame. The particle relabeling symme...