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Variational formulation for stratified steady water wave in two-layer flows

Yuchao He, Yonghui Xia, Zhe Zhou

Abstract

In this paper, the variational formulation for steady periodic stratified water waves in two-layer flows is given. The critical points of a natural energy functional is proved to be the solutions of the governing equations. And the second variation of the functional is also presented.

Variational formulation for stratified steady water wave in two-layer flows

Abstract

In this paper, the variational formulation for steady periodic stratified water waves in two-layer flows is given. The critical points of a natural energy functional is proved to be the solutions of the governing equations. And the second variation of the functional is also presented.
Paper Structure (6 sections, 3 theorems, 54 equations, 1 figure)

This paper contains 6 sections, 3 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Formulation of stratified water wave
  3. Euler equations
  4. Equivalent formulations
  5. First variational formulation
  6. Second variation

Key Result

Theorem 3.1

Let $F_i$ be defined as in F_i. Then there exist constants $p_1, p_2 \in \mathbb R$ such that, $(\Psi_1,\Psi_2, \eta,\widetilde{\eta})$ is a critical point of the functional if and only if $(\Psi_1,\Psi_2,\eta,\widetilde{\eta})$ solves psi.

Figures (1)

  • Figure 1: Two-layer stratified water wave consists of $\Omega_1$ and $\Omega_2$. The internal boundary $\widetilde{\eta}(x)$ represents the pycnoclines. As the water wave passes through $\widetilde{\eta}(x)$ from $\Omega_1$ to $\Omega_2$, the density jumps discontinuity from $\rho_1$ to $\rho_2$.

Theorems & Definitions (6)

  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Definition 4.2
  • Theorem 4.3