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Large traveling capillary-gravity waves for Darcy flow

Huy Q. Nguyen

TL;DR

This work establishes the existence of large traveling capillary-gravity waves for flows governed by Darcy's law, covering both porous media (one-phase Muskat) and vertical Hele-Shaw settings. The authors develop two reformulations: a local contraction for small waves and a Global Implicit Function Theorem (GIFT) framework that relies on invertibility of the capillary-gravity operator $\sigma H + g I$ and the Dirichlet-to-Neumann map $G[\eta]$, producing a connected solution set $\mathcal{C}$ of traveling-wave profiles. The main contributions include a rigorous construction of a local curve of small traveling waves near $(\eta,\kappa)=(0,0)$ and a global continuation result showing $\mathcal{C}$ is unbounded, with either arbitrarily large gradients or bottom-contact (finite depth), plus a regularity persistence result and a vanishing-tension limit to gravity waves. This constitutes the first nonperturbative construction of large traveling surface waves for viscous free-boundary problems and provides insight into blow-up mechanisms and the transition to gravity waves as surface tension vanishes.

Abstract

We study capillary-gravity surface waves for fluid flows governed by Darcy's law. This includes flows in vertical Hele-Shaw cells and in porous media (the one-phase Muskat problem) with finite or infinite depth. The free boundary is acted upon by an external pressure posited to be in traveling wave form with an arbitrary periodic profile and an amplitude parameter. For any given wave speed, we first prove that there exists a unique local curve of small periodic traveling waves corresponding to small values of the parameter. Then we prove that as the parameter increases but could possibly be bounded, the curve belongs to a connected set $\mathcal{C}$ of traveling waves. The set $\mathcal{C}$ contains traveling waves that either have arbitrarily large gradients or are arbitrarily close to the rigid bottom in the finite depth case. To the best of our knowledge, this is the first construction of large traveling surface waves for a viscous free boundary problem.

Large traveling capillary-gravity waves for Darcy flow

TL;DR

This work establishes the existence of large traveling capillary-gravity waves for flows governed by Darcy's law, covering both porous media (one-phase Muskat) and vertical Hele-Shaw settings. The authors develop two reformulations: a local contraction for small waves and a Global Implicit Function Theorem (GIFT) framework that relies on invertibility of the capillary-gravity operator and the Dirichlet-to-Neumann map , producing a connected solution set of traveling-wave profiles. The main contributions include a rigorous construction of a local curve of small traveling waves near and a global continuation result showing is unbounded, with either arbitrarily large gradients or bottom-contact (finite depth), plus a regularity persistence result and a vanishing-tension limit to gravity waves. This constitutes the first nonperturbative construction of large traveling surface waves for viscous free-boundary problems and provides insight into blow-up mechanisms and the transition to gravity waves as surface tension vanishes.

Abstract

We study capillary-gravity surface waves for fluid flows governed by Darcy's law. This includes flows in vertical Hele-Shaw cells and in porous media (the one-phase Muskat problem) with finite or infinite depth. The free boundary is acted upon by an external pressure posited to be in traveling wave form with an arbitrary periodic profile and an amplitude parameter. For any given wave speed, we first prove that there exists a unique local curve of small periodic traveling waves corresponding to small values of the parameter. Then we prove that as the parameter increases but could possibly be bounded, the curve belongs to a connected set of traveling waves. The set contains traveling waves that either have arbitrarily large gradients or are arbitrarily close to the rigid bottom in the finite depth case. To the best of our knowledge, this is the first construction of large traveling surface waves for a viscous free boundary problem.
Paper Structure (11 sections, 20 theorems, 167 equations)

This paper contains 11 sections, 20 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. The Dirichlet-to-Neumann operator
  3. Invertibility
  4. Linearization and contraction
  5. The capillary-gravity operator
  6. Proof of Theorem \ref{['theo:main']}
  7. Small traveling waves
  8. Large traveling waves
  9. Finite depth
  10. Infinite depth
  11. Fourier multipliers in Hölder spaces

Key Result

Theorem 1.1

We consider $\sigma>0$ and $g>0$, and assume that the fluid domain is either domain:i (infinite depth) or domain:f (finite depth) with the separation condition sep:cond. Let $d\in \{1, 2\}$ and $\alpha \in (0, 1)$. Fix an arbitrary external pressure profile $\varphi \in \mathring{C}^{1, \alpha}(\mat III. (Regularity) For any $k\ge 3$ and $\mu\in (0, 1)$, if $\varphi \in \mathring{C}^{k-2, \mu}(\ma

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 32 more