Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local Wellposedness of dispersive equations with quasi-periodic initial data

Hagen Papenburg

Abstract

We prove unconditional local well-posedness in a space of quasi-periodic functions for dispersive equations of the form $$\partial_tu + Lu + \partial_x(u^{p+1})=0,$$ where $L$ is a multiplier operator with purely imaginary symbol which grows at most exponentially. The class of equations to which our method applies includes the generalized Korteweg-de Vries equation, the generalized Benjamin-Ono equation, and the derivative nonlinear Schrödinger equation. We also discuss well-posedness of some dispersive models which do not have a problematic derivative in the nonlinearity, namely, the nonlinear Schrödinger equation and the generalized Benjamin-Bona-Mahony equation, with quasi-periodic initial data. In this way, we recover and improve upon results from arXiv:1212.2674v3 [math.AP], arXiv:2110.11263v1 [math.AP] and arXiv:2201.02920v1 [math.AP] by shorter arguments.

Local Wellposedness of dispersive equations with quasi-periodic initial data

Abstract

We prove unconditional local well-posedness in a space of quasi-periodic functions for dispersive equations of the form where is a multiplier operator with purely imaginary symbol which grows at most exponentially. The class of equations to which our method applies includes the generalized Korteweg-de Vries equation, the generalized Benjamin-Ono equation, and the derivative nonlinear Schrödinger equation. We also discuss well-posedness of some dispersive models which do not have a problematic derivative in the nonlinearity, namely, the nonlinear Schrödinger equation and the generalized Benjamin-Bona-Mahony equation, with quasi-periodic initial data. In this way, we recover and improve upon results from arXiv:1212.2674v3 [math.AP], arXiv:2110.11263v1 [math.AP] and arXiv:2201.02920v1 [math.AP] by shorter arguments.
Paper Structure (5 sections, 8 theorems, 77 equations)

This paper contains 5 sections, 8 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear and multilinear estimates
  4. Proof of Theorems \ref{['multipliertheoremreal']} and \ref{['multipliertheorem']}
  5. Models without derivative nonlinearity

Key Result

Theorem 1.1

Let $L$ be a Fourier multiplier operator with purely imaginary symbol satisfying the growth bound growthcondition and the symmetry condition symmetrycondition, and let $k>\kappa>0$. For initial data in $V_{\mathbb{R}}^{\omega,k}$, the Cauchy problem is unconditionally locally well-posed in $V_{\mathbb{R}}^{\omega,\kappa}$ both forward and backward in time.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • proof : Proof of Proposition \ref{['notionofsolution']}
  • Definition 2.3
  • Proposition 2.4: Linear estimates
  • proof
  • Remark 2.5
  • ...and 6 more