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Low regularity well-posedness for two-dimensional hydroelastic waves

Lizhe Wan, Jiaqi Yang

TL;DR

This work establishes low-regularity local well-posedness for two-dimensional irrotational deep hydroelastic waves in holomorphic coordinates by developing a paradifferential- and normal-form–based framework. Central to the approach is a cubic modified energy with a carefully chosen paradifferential weight that compensates leading nonlinear interactions, together with a hierarchy of quadratic and cubic normal-form corrections to remove non-perturbative terms. The authors prove local well-posedness in ${\mathcal H}^s$ for $s>\frac{3}{4}$ and derive refined energy estimates that close at this regularity, improving upon prior energy bounds and accommodating the high-order elastic boundary condition. The results advance the understanding of hydroelastic wave dynamics and provide a robust toolkit—para-material derivatives, paradifferential reductions, and normal-form techniques—for related quasilinear dispersive free-boundary problems with nonlinear elasticity.

Abstract

We investigate the low regularity local well-posedness of two-dimensional irrotational deep hydroelastic waves. Building on the approach of Ifrim-Tataru [29] and Ai-Ifrim-Tataru [5], in particular by constructing a cubic modified energy that incorporates a paradifferential weight chosen carefully, we prove that the hydroelastic waves are locally well-posed in $\mathcal{H}^s$ for $s>\frac{3}{4}$.

Low regularity well-posedness for two-dimensional hydroelastic waves

TL;DR

This work establishes low-regularity local well-posedness for two-dimensional irrotational deep hydroelastic waves in holomorphic coordinates by developing a paradifferential- and normal-form–based framework. Central to the approach is a cubic modified energy with a carefully chosen paradifferential weight that compensates leading nonlinear interactions, together with a hierarchy of quadratic and cubic normal-form corrections to remove non-perturbative terms. The authors prove local well-posedness in for and derive refined energy estimates that close at this regularity, improving upon prior energy bounds and accommodating the high-order elastic boundary condition. The results advance the understanding of hydroelastic wave dynamics and provide a robust toolkit—para-material derivatives, paradifferential reductions, and normal-form techniques—for related quasilinear dispersive free-boundary problems with nonlinear elasticity.

Abstract

We investigate the low regularity local well-posedness of two-dimensional irrotational deep hydroelastic waves. Building on the approach of Ifrim-Tataru [29] and Ai-Ifrim-Tataru [5], in particular by constructing a cubic modified energy that incorporates a paradifferential weight chosen carefully, we prove that the hydroelastic waves are locally well-posed in for .
Paper Structure (33 sections, 23 theorems, 299 equations, 1 figure)

This paper contains 33 sections, 23 theorems, 299 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Local well-posedness results of water waves and hydroelastic waves.
  3. Hydroelastic waves in holomorphic coordinates
  4. Main results of this paper
  5. Key Difficulties and Strategy of Proof
  6. Para-material derivatives and the linearized hydroelastic waves
  7. Leading terms of para-material derivatives
  8. The derivation of the linearized hydroelastic waves
  9. Quadratic bounds for paradifferential source terms
  10. Energy estimates for the linearized hydroelastic waves
  11. ${\mathcal{H} }^0$ energy estimate of the homogeneous paradifferential flow
  12. Normal form analysis for $({\mathbf P}\mathcal{G}_0, {\mathbf P}\mathcal{K}_0)$
  13. Low-high quadratic normal form analysis for $({\mathbf P}\mathcal{G}_0, {\mathbf P}\mathcal{K}_0)$
  14. Balanced quadratic normal form analysis for $({\mathbf P}\mathcal{G}_0, {\mathbf P}\mathcal{K}_0)$
  15. Cubic normal form analysis for $({\mathbf P}\mathcal{G}_0, {\mathbf P}\mathcal{K}_0)$
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $(\mathbf{W}, R)$ solve the hydroelastic wave system HF19. For any $s> 0$, there exists an energy functional $E_s(\mathbf{W}, R)$ that has the following properties:

Figures (1)

  • Figure 1: Hydroelastic waves

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.3
  • ...and 22 more