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Global well-posedness of the Cauchy problem for the modified Whitham equations

Han Cui, Yuexun Wang, Zhouping Xin

TL;DR

This work proves global existence and modified scattering for the Cauchy problem of the modified Whitham equation with small, smooth, localized data. The authors overcome slow decay and non-homogeneity of the dispersive symbol Λ via an interaction multiplier framework and energy estimates conducted in frequency space, complemented by a detailed resonance analysis that reveals and exploits the absence of time resonances to generate a phase correction H(ξ,t) and a scattering state w∞. A bootstrap argument combines sharp decay, Z-norm control, and weighted-norm estimates to close the estimates and establish precise long-time behavior, including a logarithmic-type phase modification quantified by H and the asymptotic convergence of e^{iH} f̂ to w∞. The results contribute to the broader understanding of long-range scattering for dispersive models with non-homogeneous dispersion and highlight the utility of frequency-space energy methods and localized multiplier techniques in managing derivative losses. The findings have potential implications for related long-range dispersive systems and the analysis of long-time dynamics in nonlocal shallow water models.

Abstract

This paper aims to show global existence and modified scattering for the solutions of the Cauchy problem to the modified Whitham equations for small, smooth and localized initial data. The main difficulties come from slow decay and non-homogeneity of the Fourier multiplier $(\sqrt{\tanh ξ/ξ})ξ$, which will be overcome by introducing an interaction multiplier theorem and estimating the weighted norms in the frequency space. When estimating the weighted norms, due to loss of derivatives, the energy estimate will be performed in the frequency space, and the absence of time resonance will be effectively utilized by extracting some good terms arising from integration by parts in time before the energy estimate.

Global well-posedness of the Cauchy problem for the modified Whitham equations

TL;DR

This work proves global existence and modified scattering for the Cauchy problem of the modified Whitham equation with small, smooth, localized data. The authors overcome slow decay and non-homogeneity of the dispersive symbol Λ via an interaction multiplier framework and energy estimates conducted in frequency space, complemented by a detailed resonance analysis that reveals and exploits the absence of time resonances to generate a phase correction H(ξ,t) and a scattering state w∞. A bootstrap argument combines sharp decay, Z-norm control, and weighted-norm estimates to close the estimates and establish precise long-time behavior, including a logarithmic-type phase modification quantified by H and the asymptotic convergence of e^{iH} f̂ to w∞. The results contribute to the broader understanding of long-range scattering for dispersive models with non-homogeneous dispersion and highlight the utility of frequency-space energy methods and localized multiplier techniques in managing derivative losses. The findings have potential implications for related long-range dispersive systems and the analysis of long-time dynamics in nonlocal shallow water models.

Abstract

This paper aims to show global existence and modified scattering for the solutions of the Cauchy problem to the modified Whitham equations for small, smooth and localized initial data. The main difficulties come from slow decay and non-homogeneity of the Fourier multiplier , which will be overcome by introducing an interaction multiplier theorem and estimating the weighted norms in the frequency space. When estimating the weighted norms, due to loss of derivatives, the energy estimate will be performed in the frequency space, and the absence of time resonance will be effectively utilized by extracting some good terms arising from integration by parts in time before the energy estimate.
Paper Structure (34 sections, 11 theorems, 376 equations)

This paper contains 34 sections, 11 theorems, 376 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Multiplier Theorems
  5. Resonance Analysis
  6. Decay Estimates
  7. Estimates for Time Derivatives
  8. Consequences of \ref{['energyasp']}-\ref{['Zasp']}
  9. $H^{N_0}$ Norms
  10. $Z$ Norms
  11. High-Low Frequency Interactions
  12. Low Frequency Interactions
  13. Estimates for Case 1
  14. Estimates for Case 2 and Case 3
  15. Estimates for Case 4
  16. ...and 19 more sections

Key Result

Theorem 1.1

Assume that $N_0=10^7$, $p_0=10^{-5}$ and $p_2=50p_0$The parameters $N_0$, $p_0$, $p_2$ could be optimized.. Suppose that for some constant $\bar{\epsilon}$ sufficiently small. Then the Cauchy problem dpb-initial admits a unique global solution $u\in C(\mathbb{R};H^{N_0}(\mathbb{R}))$ satisfying the following uniform bounds for all $t \in [0,\infty)$. Moreover, let then there exists $w_\infty\i

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 2.1
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • Corollary 1
  • proof
  • Lemma 5
  • ...and 5 more