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Propagation of singularities for gravity-capillary water waves

Hui Zhu

Abstract

We obtain two results of propagation for solutions to the gravity-capillary water wave system. First we show how oscillations and the spatial decay propagate at infinity; then we show a microlocal smoothing effect under the non-trapping condition of the initial free surface. These results extends the works of Craig, Kappeler and Strauss, Wunsch and Nakamura to quasilinear dispersive equations. We also prove the existence of gravity-capillary water waves in weighted Sobolev spaces. Such solutions have asymptotically Euclidean free surfaces. To obtain these results, we generalize the paradifferential calculus of Bony to weighted Sobolev spaces and develop a semiclassical paradifferential calculus. We also introduce a new family of wavefront sets -- the quasi-homogeneous wavefront sets, which is a generalization, at least in the Euclidean geometry, the wavefront sets of Hörmander, the scattering wavefront sets of Melrose, the quadratic scattering wavefront sets of Wunsch and the homogeneous wavefront sets of Nakamura.

Propagation of singularities for gravity-capillary water waves

Abstract

We obtain two results of propagation for solutions to the gravity-capillary water wave system. First we show how oscillations and the spatial decay propagate at infinity; then we show a microlocal smoothing effect under the non-trapping condition of the initial free surface. These results extends the works of Craig, Kappeler and Strauss, Wunsch and Nakamura to quasilinear dispersive equations. We also prove the existence of gravity-capillary water waves in weighted Sobolev spaces. Such solutions have asymptotically Euclidean free surfaces. To obtain these results, we generalize the paradifferential calculus of Bony to weighted Sobolev spaces and develop a semiclassical paradifferential calculus. We also introduce a new family of wavefront sets -- the quasi-homogeneous wavefront sets, which is a generalization, at least in the Euclidean geometry, the wavefront sets of Hörmander, the scattering wavefront sets of Melrose, the quadratic scattering wavefront sets of Wunsch and the homogeneous wavefront sets of Nakamura.

Paper Structure

This paper contains 50 sections, 70 theorems, 465 equations.

Table of Contents

  1. Introduction
  2. Wavefront set and the linear half wave equation
  3. Homogeneous wavefront set and the linear Schrödinger equation
  4. Quasi-homogeneous wavefront set and the gravity-capillary water wave system
  5. Formulations of the gravity-capillary water wave system
  6. Quasi-homogeneous wavefront set and model equations
  7. Asymptotically flat water waves
  8. Propagation at infinity
  9. Microlocal smoothing effect
  10. Outline of paper
  11. Quasi-homogeneous microlocal analysis
  12. Quasi-homogeneous semiclassical calculus
  13. Weighted Sobolev spaces
  14. The quasi-homogeneous wavefront set
  15. Model equations
  16. ...and 35 more sections

Key Result

Theorem 1.1

Let $M$ be a smooth manifold without boundary. Let $P \in \Psi^1(M)$ which admits a real principal symbol $\sigma(P) = \sigma(P)(x,\xi) \in C^\infty(T^*M \backslash 0,\mathbb{R})$ and let $\Phi = \Phi_t(x,\xi) \in C^\infty(\mathbb{R}\times T^*M \backslash 0,T^*M \backslash 0)$ be the Hamiltonian flo then for all $(x_0,\xi_0) \in \mathrm{WF}(u_0)$ and all $t \in \mathbb{R}$, we have $\Phi_t(x_0,\xi

Theorems & Definitions (141)

  • Theorem 1.1: Hörmander Hormander71fourier
  • Theorem 1.2: Nakamura Nakamura05propagation, similar results by Wunsch Wunsch99propagation
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 131 more