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The Benjamin-Ono equation with quasi-periodic data

Hagen Papenburg

TL;DR

The paper analyzes the Benjamin-Ono equation with quasi-periodic data and proves local well posedness for small data in the Yomega_sigma space with sigma>7/8. It combines a Strichartz based a priori bound for quasi periodic Strichartz estimates, derived via decoupling, with a quasi periodic gauge transform to control nonlinear interactions in a setting where dispersion is degenerate. The approach relies on anisotropic Sobolev spaces and intricate paraproduct and commutator estimates to establish uniform bounds, Cauchy limits, and continuity of the data-to-solution map, culminating in a robust local theory for quasi periodic data and a Diophantine based embedding that broadens the admissible initial data. The results advance the understanding of dispersive equations under quasi periodic conditions and provide a framework potentially adaptable to other models such as KdV and dNLS.

Abstract

We construct local solutions to the Benjamin-Ono equation for quasi-periodic initial data. The solution is unique among limits of smooth solutions and depends continuously on the data. Our result applies to a richer class of quasi-periodic functions than previous theorems. Central to the argument is an a-priori estimate, the proof of which utilizes Strichartz estimates for quasi-periodic functions obtained recently via decoupling, and a quasi-periodic extension of Tao's gauge transform. As a byproduct of our method, we also establish new local wellposedness results in certain anisotropic Sobolev spaces.

The Benjamin-Ono equation with quasi-periodic data

TL;DR

The paper analyzes the Benjamin-Ono equation with quasi-periodic data and proves local well posedness for small data in the Yomega_sigma space with sigma>7/8. It combines a Strichartz based a priori bound for quasi periodic Strichartz estimates, derived via decoupling, with a quasi periodic gauge transform to control nonlinear interactions in a setting where dispersion is degenerate. The approach relies on anisotropic Sobolev spaces and intricate paraproduct and commutator estimates to establish uniform bounds, Cauchy limits, and continuity of the data-to-solution map, culminating in a robust local theory for quasi periodic data and a Diophantine based embedding that broadens the admissible initial data. The results advance the understanding of dispersive equations under quasi periodic conditions and provide a framework potentially adaptable to other models such as KdV and dNLS.

Abstract

We construct local solutions to the Benjamin-Ono equation for quasi-periodic initial data. The solution is unique among limits of smooth solutions and depends continuously on the data. Our result applies to a richer class of quasi-periodic functions than previous theorems. Central to the argument is an a-priori estimate, the proof of which utilizes Strichartz estimates for quasi-periodic functions obtained recently via decoupling, and a quasi-periodic extension of Tao's gauge transform. As a byproduct of our method, we also establish new local wellposedness results in certain anisotropic Sobolev spaces.

Paper Structure

This paper contains 27 sections, 52 theorems, 374 equations.

Table of Contents

  1. introduction
  2. Organization of paper
  3. Acknowledgements
  4. notation and basic properties
  5. preliminaries
  6. Properties of anisotropic Sobolev spaces
  7. Tools for dispersive PDE in the quasi-periodic setting: commutator estimates
  8. wellposedness in anisotropic Sobolev spaces and in $X_{\omega}^{s_1,s_2}$
  9. Regularized equation
  10. Uniform a-priori estimate
  11. Cauchy bound
  12. Solution to Benjamin--Ono equation
  13. Uniqueness
  14. Continuity of data-to-solution map
  15. Wellposedness in $X$
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $\sigma>7/8$, $\omega \in \mathbb{R}^2$ and denote $Y:=Y^{\sigma}_{\omega}$ as in Definition Yspace. There exists an absolute constant $\rho_0$ such that for any initial data $u_0 \in B_{Y}(0,\rho_0)$ there exists a solution to the Benjamin--Ono equation in $C([0,1],H^{1+\tilde{\sigma},\tilde{\s

Theorems & Definitions (101)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3: Anisotropic Sobolev spaces
  • Theorem 1.4
  • Theorem 1.5: a-priori bound
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 1.7: growth bound for $X$-norm
  • Remark 1.8
  • ...and 91 more