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Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data

Laurent Lafleche

Abstract

We prove a quantitative and global in time semiclassical limit from the Hartree to the Vlasov equation in the case of a singular interaction potential in dimension $d\geq 3$, including the case of a Coulomb singularity in dimension $d=3$. This result holds for initial data concentrated enough in the sense that some space moments are initially sufficiently small. As an intermediate result, we also obtain quantitative semiclassical bounds on the space and velocity moments of even order and the asymptotic behavior of the spatial density due to dispersion effects.

Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data

Abstract

We prove a quantitative and global in time semiclassical limit from the Hartree to the Vlasov equation in the case of a singular interaction potential in dimension , including the case of a Coulomb singularity in dimension . This result holds for initial data concentrated enough in the sense that some space moments are initially sufficiently small. As an intermediate result, we also obtain quantitative semiclassical bounds on the space and velocity moments of even order and the asymptotic behavior of the spatial density due to dispersion effects.

Paper Structure

This paper contains 5 sections, 10 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Free Transport
  4. Propagation of moments
  5. Classical case.

Key Result

Theorem 1

Let $d\geq 3$, $n\in 2\mathbb N\setminus\{0\}$, $r\in[1,\infty]$ and define $\mathfrak b_n := \frac{nr'+d}{n+1}$. Assume and let $\boldsymbol{\rho}$ be a solution of the eq:Hartree equation with initial condition Then there exists an explicit constant $\mathcal{C}>0$ depending on $M_0$, $\mathop{\mathrm{Tr}}\nolimits(\left|{\boldsymbol{p}}\right|^n\boldsymbol{\rho}^\mathrm{in})$, $\left\|{\nabla

Theorems & Definitions (19)

  • Theorem 1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 1.1
  • Remark 1.5
  • Theorem 2
  • Proposition 1.2
  • Theorem 3: Golse & Paul golse_schrodinger_2017
  • ...and 9 more