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Modified Wave operators for the Hartree equation with repulsive Coulomb potential

Wenrui Huang

Abstract

We study the final state problem for the Hartree equation with repulsive Coulomb potential: \[i\partial_t u+\frac{1}{2}Δu-\frac{1}{|x|}u=((-Δ)^{-1}|u|)^2u\] We show the work in \cite{KaMi} can be extended to the Hartree nonlinearity: Given a prescribed asymptotic profile, we construct a unique global solution scattering to the profile. In particular, the existence of the modified wave operators is obtained for sufficiently localized small scattering data.

Modified Wave operators for the Hartree equation with repulsive Coulomb potential

Abstract

We study the final state problem for the Hartree equation with repulsive Coulomb potential: We show the work in \cite{KaMi} can be extended to the Hartree nonlinearity: Given a prescribed asymptotic profile, we construct a unique global solution scattering to the profile. In particular, the existence of the modified wave operators is obtained for sufficiently localized small scattering data.

Paper Structure

This paper contains 9 sections, 11 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Notations and definitions
  4. Strichartz Estimates
  5. Construction of the asymptotic profile $u_p$
  6. Proof of the main theorem
  7. Reformulation into a fixed point argument
  8. Estimates of $\mathcal{K}[u]$
  9. Control the error term $\mathcal{E}(t)$

Key Result

Theorem 1.1

Suppose $\frac{1}{4}<b<\frac{1}{2}$, $\frac{2}{q}+\frac{3}{r}=\frac{3}{2}$ , $\widehat{u_+}\in H^1(\mathbb{R}^3)$ and $0\notin \mathrm{supp}\; \widehat{u_{+}}$ and $\|(-\Delta)^{-1}|\widehat{u_{+}}|^2 \|_{L^\infty}$ be small enough. Then there exists a unique solution $u\in C(\mathbb{R};L^2(\mathbb{

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4: Modified wave operator
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3: Nonlinear Estimates
  • proof
  • Theorem 2.4: Global-in-time Strichartz Estimates Mizutani
  • ...and 14 more