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Bifurcation analysis of Stokes waves with piecewise smooth vorticity in deep water

Changfeng Gui, Jun Wang, Wen Yang, Yong Zhang

TL;DR

This work proves the existence of large-amplitude two-dimensional periodic Stokes waves in deep water with piecewise smooth vorticity across a single interface. By combining the hodograph transformation, the height-function framework, and a transmission formulation, the authors convert the free-boundary problem into an elliptic system with an internal jump, then employ a singular global bifurcation argument via Whyburn's lemma. An approximating family $F^{oldsymbol{ extvarepsilon}}$ yields Fredholm linearizations and a global bifurcation structure, which is carefully passed to the limit $oldsymbol{ extvarepsilon} o 0$ to obtain a continuum of nontrivial solutions of the original problem. The results demonstrate that along the global branch either the wave speed becomes unbounded or horizontal stagnation is approached, providing a rigorous mechanism for large-amplitude, piecewise-rotational deep-water waves with internal interfaces.

Abstract

In this paper, we establish the existence of Stokes waves with piecewise smooth vorticity in a two-dimensional, infinitely deep fluid domain. These waves represent traveling water waves propagating over sheared currents in a semi-infinite cylinder, where the vorticity may exhibit discontinuities. The analysis is carried out by applying a hodograph transformation, which reformulates the original free boundary problem into an abstract elliptic boundary value problem. Compared to previously studied steady water waves, the present setting introduces several novel features: the presence of an internal interface, an unbounded spatial domain, and a non-Fredholm linearized operator. To address these difficulties, we introduce a height function formulation, casting the problem as a transmission problem with suitable transmission conditions. A singular bifurcation approach is then employed, combining global bifurcation theory with Whyburns topological lemma. Along the global bifurcation branch, we show that the resulting wave profiles either attain arbitrarily large wave speed or approach horizontal stagnation.

Bifurcation analysis of Stokes waves with piecewise smooth vorticity in deep water

TL;DR

This work proves the existence of large-amplitude two-dimensional periodic Stokes waves in deep water with piecewise smooth vorticity across a single interface. By combining the hodograph transformation, the height-function framework, and a transmission formulation, the authors convert the free-boundary problem into an elliptic system with an internal jump, then employ a singular global bifurcation argument via Whyburn's lemma. An approximating family yields Fredholm linearizations and a global bifurcation structure, which is carefully passed to the limit to obtain a continuum of nontrivial solutions of the original problem. The results demonstrate that along the global branch either the wave speed becomes unbounded or horizontal stagnation is approached, providing a rigorous mechanism for large-amplitude, piecewise-rotational deep-water waves with internal interfaces.

Abstract

In this paper, we establish the existence of Stokes waves with piecewise smooth vorticity in a two-dimensional, infinitely deep fluid domain. These waves represent traveling water waves propagating over sheared currents in a semi-infinite cylinder, where the vorticity may exhibit discontinuities. The analysis is carried out by applying a hodograph transformation, which reformulates the original free boundary problem into an abstract elliptic boundary value problem. Compared to previously studied steady water waves, the present setting introduces several novel features: the presence of an internal interface, an unbounded spatial domain, and a non-Fredholm linearized operator. To address these difficulties, we introduce a height function formulation, casting the problem as a transmission problem with suitable transmission conditions. A singular bifurcation approach is then employed, combining global bifurcation theory with Whyburns topological lemma. Along the global bifurcation branch, we show that the resulting wave profiles either attain arbitrarily large wave speed or approach horizontal stagnation.

Paper Structure

This paper contains 14 sections, 18 theorems, 181 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The historical background
  3. The plan of the paper
  4. Equivalent formulations and main results
  5. Governing equations in velocity field formulation
  6. Governing equation in stream function formulation
  7. Governing equation in height function formulation
  8. Main results
  9. The related transmission problem and laminar flow
  10. The bifurcation of the approximating problem
  11. The global bifurcation of the approximating problem
  12. The bifurcation structure of the approximating problem
  13. Global existence of Stokes waves with piecewise smooth vorticity
  14. Proof of Theorem \ref{['thm2.1']}

Key Result

Theorem 2.1

Suppose that the vorticity function $\gamma\in C^{1,\alpha}([0,-p_0))\cap C^{1,\alpha}([-p_0, \infty))$ with $\alpha\in (0,1)$, satisfies $\gamma(s) \in O(s^{-2-r})$ for $r>0$ as $s\rightarrow \infty$ and $-\Gamma_{inf}<\frac{g^{\frac{2}{3}}}{4}$. Then there exists a connected set $\mathcal{K}$ in t

Figures (3)

  • Figure 1: The schematic of the problem.
  • Figure 2: The profile of the eigenvalue $\mu^{\varepsilon}(\lambda)$.
  • Figure 3: The nodal domain.

Theorems & Definitions (32)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • ...and 22 more