A comparison theorem for super- and subsolutions of $\mathbf{\nabla^2 u + f (u) = 0}$ and its application to water waves with vorticity

TL;DR

This paper establishes a comparison theorem for solutions of a nonlinear PDE with applications to steady water waves with vorticity, providing bounds on wave profiles and flow parameters.

Contribution

It introduces a new comparison theorem for solutions with nonlinearities in $L^p_{loc}$ and applies it to complex water wave problems with vorticity.

Findings

Derived bounds for free-surface profiles including overhanging waves

Established limits for streamfunctions and total head values

Extended analysis to arbitrary wave profiles

Abstract

A comparison theorem is proved for a pair of solutions that satisfy in a weak sense opposite differential inequalities with nonlinearity of the form $f (u)$ with $f$ belonging to the class $L^p_{loc}$. The solutions are assumed to have non-vanishing gradients in the domain, where the inequalities are considered. The comparison theorem is applied to the problem describing steady, periodic water waves with vorticity in the case of arbitrary free-surface profiles including overhanging ones. Bounds for these profiles as well as streamfunctions and admissible values of the total head are obtained.