Monokinetic solutions to a singular Vlasov equation from a semiclassical perspective

TL;DR

This paper investigates monokinetic solutions to a singular 1D Vlasov equation by analyzing the semiclassical limit of a related logarithmic Schrödinger equation, providing insights into global solutions and their relation to Euler systems.

Contribution

It introduces a novel approach linking the semiclassical limit of a logarithmic Schrödinger equation to solutions of a singular Vlasov equation, including global solutions to the isothermal Euler system.

Findings

Derived monokinetic solutions from the semiclassical limit.

Established global solutions to the isothermal Euler system.

Analyzed initial densities, including Gaussian and smooth away from vacuum.

Abstract

Solutions to a singular one-dimensional Vlasov equation are obtained as the semiclassical limit of the Wigner transform associated to a logarithmic Schrodinger equation. Two frameworks are considered, regarding in particular the initial position density: Gaussian initial density, or smooth initial density away from vacuum. For Gaussian initial densities, the analysis also yields global solutions to the isothermal Euler system that do not enter the frame of regular solutions to hyperbolic systems by P. D. Lax.