On large periodic traveling surface waves in porous media

TL;DR

This paper constructs large traveling surface waves in a viscous porous fluid without surface tension, using a novel reformulation and ellipticity discovery to prove existence of solutions with arbitrarily large amplitude.

Contribution

It introduces a non perturbative method to establish the existence of large amplitude traveling waves in viscous porous media without surface tension, a first in the field.

Findings

Existence of a connected set of traveling wave solutions.

Solutions can exist for arbitrarily large data amplitude.

Finite breakdown scenarios are identified for large solutions.

Abstract

We study large traveling surface waves within a two-dimensional finite depth, free boundary, homogeneous, incompressible and viscous fluid governed by Darcy's law. The fluid is bound by a gravitational force to a flat rigid bottom and meets an atmosphere of constant pressure at the top with its free surface, where it does not experience any capillarity effects. Additionally, the fluid is subject to a fixed, but arbitrarily selected, forcing data profile with variable amplitude. We use the Riemann mapping to equivalently reformulate the resulting two-dimensional free boundary problem as a single one-dimensional fully nonlinear pseudodifferential equation for a function describing the domain's geometry. By discovering a hidden ellipticity in the reformulated equation, we are able to import a global implicit function theorem to construct a connected set of traveling waves, containing both the quiescent solution and large amplitude members. We find that either solutions continue to exist for arbitrarily large data amplitude or else one of a finite number of meaningful breakdown scenarios must occur. This work stands as the first non perturbative construction of large traveling surface waves in any free boundary viscous fluid without surface tension.