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Weakly nonlinear models for hydroelastic water waves

Diego Alonso-Orán, Rafael Granero-Belinchón, Juliana S. Ziebell

Abstract

In this work, we derive reduced interface models for hydroelastic water waves coupled to a nonlinear viscoelastic plate. In a weakly nonlinear small-steepness regime we obtain bidirectional nonlocal evolution equations capturing the interface dynamics up to quadratic order, and we also derive two unidirectional models describing one-way propagation while retaining the leading dispersive and dissipative effects induced by the plate. Remarkably, one of the bidirectional model has a doubly nonlinear structure in the sense that there there is a nonlinear elliptic operator acting on the acceleration of the interface. We prove local well-posedness for the bidirectional model for small data via a two-parameter regularization and nested fixed points. For the unidirectional models, we obtain local well-posedness for arbitrary data and global well-posedness for small data.

Weakly nonlinear models for hydroelastic water waves

Abstract

In this work, we derive reduced interface models for hydroelastic water waves coupled to a nonlinear viscoelastic plate. In a weakly nonlinear small-steepness regime we obtain bidirectional nonlocal evolution equations capturing the interface dynamics up to quadratic order, and we also derive two unidirectional models describing one-way propagation while retaining the leading dispersive and dissipative effects induced by the plate. Remarkably, one of the bidirectional model has a doubly nonlinear structure in the sense that there there is a nonlinear elliptic operator acting on the acceleration of the interface. We prove local well-posedness for the bidirectional model for small data via a two-parameter regularization and nested fixed points. For the unidirectional models, we obtain local well-posedness for arbitrary data and global well-posedness for small data.

Paper Structure

This paper contains 22 sections, 5 theorems, 376 equations.

Table of Contents

  1. Introduction
  2. Contributions and main results
  3. Notation and preliminary estimates
  4. Organization of the manuscript
  5. Setting of the problem
  6. The fluid structure interaction system
  7. Boundary conditions, geometric bending energy and structural forces
  8. The full Euler–plate system and dimensionless system
  9. The Arbitrary Lagrangian-Eulerian (ALE) formulation
  10. Formal asymptotic expansion
  11. The unidirectional models
  12. The unidirectional model for \ref{['eq:f_model_compact_final_again']}
  13. Well-posedness result for the bidirectional model
  14. Well-posedness results for the unidirectional models
  15. Geometric formulas for graph surfaces
  16. ...and 7 more sections

Key Result

Lemma 1.1

Let $b=b(x,x_3)$ and $g=g(x)$ be smooth, $2\pi$--periodic in $x\in\mathbb T^2$, and assume that $b(\cdot,x_3)$ decays sufficiently fast as $x_3\to-\infty$. Consider the problem Then eq:Poisson_halfspace_DN admits a unique solution $u$, and for every nonzero Fourier mode $k\in\mathbb Z^2\setminus\{0\}$ one has Equivalently, in operator form on mean--zero functions,

Theorems & Definitions (17)

  • Lemma 1.1: Poisson problem with Dirichlet data
  • proof
  • Remark 2.1: General bending energies
  • Remark 2.2: Dependence on the Gauss curvature
  • Remark 2.3: Leading--order reduction of the bending operator
  • Theorem 4.1: Local well-posedness for small $H^3$ data
  • proof : Proof of Theorem \ref{['thm:WPbi']}
  • Remark 4.2
  • Theorem 5.1: Local well-posedness
  • proof : Proof of Theorem \ref{['th:LWP:uni']}
  • ...and 7 more