On the $L^2$ Rate of Convergence in the Limit from the Hartree to the Vlasov$\unicode{x2013}$Poisson Equation

Abstract

Using a new stability estimate for the difference of the square roots of two solutions of the Vlasov$\unicode{x2013}$Poisson equation, we obtain the convergence in the $L^2$ norm of the Wigner transform of a solution of the Hartree equation with Coulomb potential to a solution of the Vlasov$\unicode{x2013}$Poisson equation, with a rate of convergence proportional to $\hslash$. This improves the ${\hslash }^{3/4-\epsilon }$ rate of convergence in $L^2$ obtained in [L.~Lafleche, C.~Saffirio: Analysis & PDE, to appear]. Another reason of interest of this paper is the new method, reminiscent of the ones used to prove the mean-field limit from the many-body Schrödinger equation towards the Hartree$\unicode{x2013}$Fock equation for mixed states.

21 pages

References in corpus (5)

• Semiclassical limit to the Vlasov equation with inverse power law potentials
• From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials
• Strong semiclassical limit from Hartree and Hartree-Fock to Vlasov-Poisson equation
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Cited by in corpus (5)

• On Quantum Sobolev Inequalities
• Optimal Semiclassical Regularity of Projection Operators and Strong Weyl Law
• Semi-classical limit of the massive Klein-Gordon-Maxwell system toward the relativistic Euler-Maxwell system via an adapted modulated energy method
• Uniqueness criteria for the Vlasov--Poisson system and applications to semiclassical analysis
• Enhanced Stability in Quantum Optimal Transport Pseudometrics: From Hartree to Vlasov-Poisson