Gravity driven traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations

TL;DR

This paper constructs the first mathematical two-dimensional traveling bore wave solutions for free boundary incompressible Navier-Stokes equations in a finite depth layer, using a shallow water limit and fixed point methods.

Contribution

It provides a rigorous mathematical construction of bore wave solutions by justifying the shallow water approximation and solving the resulting ODEs.

Findings

First mathematical construction of bore wave solutions

Validation of the shallow water limit for free boundary Navier-Stokes

Use of fixed point and perturbation methods in thin domains

Abstract

We give the first mathematical construction of two-dimensional traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations for a single finite depth layer of constant density fluid. Our construction is based on a rigorous justification of the formal shallow water limit, which postulates that in a certain scaling regime the full free boundary traveling Navier-Stokes system of PDEs reduces to a governing system of ODEs. We find heteroclinic orbits solving these ODEs and, through a delicate fixed point argument employing the Stokes problem in thin domains and a nonautonomous orbital perturbation theory, use these ODE solutions as the germs from which we build bore PDE solutions for sufficiently shallow layers.