Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the capillary water waves with constant vorticity

Lizhe Wan

Abstract

This article is devoted to the study of local well-posedness for deep water waves with constant vorticity in two space dimensions on the real line. The water waves can be paralinearized and written as a quasilinear dispersive system of equations. By using the energy estimate and the Strichartz estimate, we show that for $s> \frac{5}{4}$, the gravity-capillary water wave system with constant vorticity is locally well-posed in $\mathcal{H}^{s}(\mathbb{R})$.

On the capillary water waves with constant vorticity

Abstract

This article is devoted to the study of local well-posedness for deep water waves with constant vorticity in two space dimensions on the real line. The water waves can be paralinearized and written as a quasilinear dispersive system of equations. By using the energy estimate and the Strichartz estimate, we show that for , the gravity-capillary water wave system with constant vorticity is locally well-posed in .

Paper Structure

This paper contains 20 sections, 30 theorems, 253 equations.

Table of Contents

  1. Introduction
  2. Water waves in holomorphic coordinates
  3. Some previous results
  4. The main results
  5. Paradifferential estimates and water waves related bounds
  6. Norms and function spaces
  7. Paradifferential estimates
  8. Water waves related bounds
  9. Paralinearization and reduction of the water wave system
  10. Paralinearization of the water wave system
  11. Reduction of the water wave system
  12. Symmetrization of the water wave system
  13. A priori energy estimate and the Strichartz estimate
  14. Energy estimate
  15. The Strichartz estimate
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $s> 0$. Suppose $({\mathbf W}, R)$ solve the water wave system e:WR on $[0, T]$, then we have the energy estimate for any $t\in [0,T]$,

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2: Symbolic calculus, MR2418072
  • Lemma 2.3: MR3260858
  • Lemma 2.4: MR3585049
  • Lemma 2.5: MR2768550
  • Lemma 2.6: MR2768550
  • Lemma 2.7: Paralinearization MR3770970
  • ...and 38 more