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Nonlinear smoothing for the periodic dispersion generalized Benjamin-Ono equations with polynomial nonlinearity

Wangseok Shin

Abstract

We consider the periodic dispersion generalized Benjamin-Ono equations with polynomial nonlinearity. We establish the nonlinear smoothing properties of these equations, according to which the difference between the solution and the linear evolution is smoother than the initial data. In addition, we establish new local well-posedness results for these equations when the dispersion is sufficiently large. Our method also improves known local well-posedness results for a class of non-integrable fifth-order KdV equations.

Nonlinear smoothing for the periodic dispersion generalized Benjamin-Ono equations with polynomial nonlinearity

Abstract

We consider the periodic dispersion generalized Benjamin-Ono equations with polynomial nonlinearity. We establish the nonlinear smoothing properties of these equations, according to which the difference between the solution and the linear evolution is smoother than the initial data. In addition, we establish new local well-posedness results for these equations when the dispersion is sufficiently large. Our method also improves known local well-posedness results for a class of non-integrable fifth-order KdV equations.

Paper Structure

This paper contains 34 sections, 41 theorems, 362 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Basic notations
  4. Comments on the proof
  5. Outline of the paper
  6. Analysis of the resonance functions
  7. Normal form reduction
  8. The gauged equation
  9. Decomposition of the frequency domain
  10. Differentiation by parts
  11. Preliminary lemmas
  12. Pointwise estimates
  13. Smoothing estimates
  14. Multilinear estimates
  15. Boundary terms
  16. ...and 19 more sections

Key Result

Theorem 1.1

For $1 < \alpha < 2$ and $d \geq 2$, let For $s> s(d,\alpha)$, let $u \in C([-T,T],H^{s})$ be a mean-zero solution to the equation (eq: dgbo) with $\textnormal{deg}(P)=d$. Then we have for any $0\leq a < a(d,\alpha)$ with $a \leq \alpha-1$.

Theorems & Definitions (83)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 73 more