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On large periodic traveling surface waves in porous media

Huy Q. Nguyen, Noah Stevenson

TL;DR

The paper addresses large traveling surface waves in a 2D finite-depth porous medium governed by Darcy's law, without surface tension, by recasting the free-boundary problem into a 1D fully nonlinear pseudodifferential equation for a domain-coordinate function $ heta$ via a conformal flattening (Riemann mapping) and a Dirichlet-to-Neumann framework. A hidden ellipticity structure is identified, enabling a global implicit-function/Brouwer-type continuation (Leray–Schauder) to produce a connected set of traveling-wave solutions emanating from the trivial state; along this set, either the amplitude or a geometric distortion quantity must blow up, signaling potential breakdown. The authors then rigorously transfer these solutions back to the original free-boundary Darcy problem, showing that large traveling waves exist and may be graphical or undergo conformal degeneracy. The work establishes the first non-perturbative construction of large traveling surface waves in free-boundary viscous Darcy flow without surface tension, and provides a framework for understanding global solution branches and possible breakdown scenarios in porous-media wave dynamics.

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.

On large periodic traveling surface waves in porous media

TL;DR

The paper addresses large traveling surface waves in a 2D finite-depth porous medium governed by Darcy's law, without surface tension, by recasting the free-boundary problem into a 1D fully nonlinear pseudodifferential equation for a domain-coordinate function via a conformal flattening (Riemann mapping) and a Dirichlet-to-Neumann framework. A hidden ellipticity structure is identified, enabling a global implicit-function/Brouwer-type continuation (Leray–Schauder) to produce a connected set of traveling-wave solutions emanating from the trivial state; along this set, either the amplitude or a geometric distortion quantity must blow up, signaling potential breakdown. The authors then rigorously transfer these solutions back to the original free-boundary Darcy problem, showing that large traveling waves exist and may be graphical or undergo conformal degeneracy. The work establishes the first non-perturbative construction of large traveling surface waves in free-boundary viscous Darcy flow without surface tension, and provides a framework for understanding global solution branches and possible breakdown scenarios in porous-media wave dynamics.

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.
Paper Structure (15 sections, 12 theorems, 153 equations)

This paper contains 15 sections, 12 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Equations of motion and traveling ansatz
  3. Literature Survey
  4. Nondimensional, conformal, and nonlocal reformulations
  5. Main results and discussion
  6. Conventions of notation
  7. Preliminaries
  8. Technical estimates and mapping properties
  9. Instantiation of the nonlinear operator and decompositions
  10. Analysis of large solutions to the nonlinear operator equation
  11. Construction of the connected set of solutions
  12. Improvements on the blow-up quantity
  13. Further refinements
  14. Conformal degeneracy
  15. Synthesis

Key Result

Lemma 2.1

For any $\mathbb{R}\ni s\geqslant1/2$, $\upiota\in\{{0,1}\}$, and $0<\alpha<1$ the linear maps are bounded. Moreover, for all $\psi$ belonging to one of the domains of the above maps we have that $\mathcal{E}\psi$ is smooth in the interior and solves the Cauchy-Riemann equations along with the boundary conditions

Theorems & Definitions (26)

  • Lemma 2.1: Boundedness of the Cauchy-Riemann solver
  • proof
  • Lemma 2.2: Mapping properties of the composition operators
  • proof
  • Lemma 2.3: Analysis of the lower-order bulk remainder
  • proof
  • Proposition 2.4: Commutator estimates for $\mathbb{P}_\pm$
  • proof
  • Lemma 2.5: Operator encoding
  • proof
  • ...and 16 more