Traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations

We discuss the first construction of traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations. Our construction is based on a rigorous justification of the shallow water limit, which postulates that in a certain 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 shallow layers.