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On large periodic traveling wave solutions to the free boundary Stokes and Navier-Stokes equations

Seyed Abdolhamid Banihashemi, Huy Q. Nguyen

TL;DR

The paper develops a rigorous framework for large periodic traveling waves in free boundary Stokes and Navier–Stokes fluids with surface tension and traveling external stress. By introducing and exploiting nonlocal normal-stress to normal-Dirichlet operators Ψ_γ[η] and Φ_γ[η], it reduces the traveling-wave problem to a fixed-point equation in Sobolev spaces and proves existence of large waves via contraction mappings, with γ small and stress strength large. It further establishes asymptotic stability for traveling waves in the Stokes setting, using coercivity and commutator estimates, energy methods, and a careful linear-nonlinear decomposition; this provides a clear contrast to inviscid waves and extends prior Darcy-flow results to viscous fluids. The results rely on a non-flat-domain Stokes/NS theory, sharp Sobolev regularity via domain flattening, and paralinearization techniques, with implications for understanding how external traveling forces can sustain large viscous waves.

Abstract

We study the free boundary problem for a finite-depth layer of viscous incompressible fluid in arbitrary dimension, modeled by the Stokes or Navier-Stokes equations. In addition to the gravitational field acting in the bulk, the free boundary is acted upon by surface tension and an external stress tensor posited to be in traveling wave form. We prove that for any isotropic stress tensor with periodic profile, there exists a locally unique periodic traveling wave solution, which can have large amplitude. Moreover, we prove that the constructed traveling wave solutions are asymptotically stable for the dynamic free boundary Stokes equations. Our proofs rest on the analysis of the nonlocal normal-stress to normal-Dirichlet operators for the Stokes and Navier-Stokes equations in domains of Sobolev regularity.

On large periodic traveling wave solutions to the free boundary Stokes and Navier-Stokes equations

TL;DR

The paper develops a rigorous framework for large periodic traveling waves in free boundary Stokes and Navier–Stokes fluids with surface tension and traveling external stress. By introducing and exploiting nonlocal normal-stress to normal-Dirichlet operators Ψ_γ[η] and Φ_γ[η], it reduces the traveling-wave problem to a fixed-point equation in Sobolev spaces and proves existence of large waves via contraction mappings, with γ small and stress strength large. It further establishes asymptotic stability for traveling waves in the Stokes setting, using coercivity and commutator estimates, energy methods, and a careful linear-nonlinear decomposition; this provides a clear contrast to inviscid waves and extends prior Darcy-flow results to viscous fluids. The results rely on a non-flat-domain Stokes/NS theory, sharp Sobolev regularity via domain flattening, and paralinearization techniques, with implications for understanding how external traveling forces can sustain large viscous waves.

Abstract

We study the free boundary problem for a finite-depth layer of viscous incompressible fluid in arbitrary dimension, modeled by the Stokes or Navier-Stokes equations. In addition to the gravitational field acting in the bulk, the free boundary is acted upon by surface tension and an external stress tensor posited to be in traveling wave form. We prove that for any isotropic stress tensor with periodic profile, there exists a locally unique periodic traveling wave solution, which can have large amplitude. Moreover, we prove that the constructed traveling wave solutions are asymptotically stable for the dynamic free boundary Stokes equations. Our proofs rest on the analysis of the nonlocal normal-stress to normal-Dirichlet operators for the Stokes and Navier-Stokes equations in domains of Sobolev regularity.
Paper Structure (26 sections, 42 theorems, 436 equations)

This paper contains 26 sections, 42 theorems, 436 equations.

Table of Contents

  1. Introduction
  2. The free boundary Stokes and Navier-Stokes equations
  3. Main results
  4. Method of proof
  5. Construction of large traveling waves
  6. Stability of large traveling waves
  7. Organization of the paper
  8. $\gamma-$Stokes problems in non-flat domains
  9. Weak solutions
  10. Regularity of weak solutions
  11. Normal-stress to normal-Dirichlet operators
  12. The linear normal-stress to normal-Dirichlet operator $\Psi_\gamma[\eta]$
  13. The nonlinear normal-stress to normal-Dirichlet operator $\Phi_\gamma[\eta]$
  14. Further properties of $\Psi_\gamma[\eta]$
  15. Coercive estimates
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $(d+1)/2<s\in \mathbb{N}$, and assume that $\phi\in H^{s-\frac{1}{2}}(\mathbb{T}^d)\cap C^{1}(\mathbb{T}^d)$ with $\min_{\mathbb{T}^d} (-\phi)>-g b$. Then there exist $\delta_0=\delta_0(\| \phi\|_{H^{s-\frac{1}{2}}\cap C^{1}})>0$ such that for any $\delta\in (0, \delta_0)$, there is some $\gamm

Theorems & Definitions (85)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • ...and 75 more