Local analytic well-posedness for one-dimensional Vlasov$\unicode{x2013}$Dirac$\unicode{x2013}$Benney-type equations

TL;DR

This paper proves local well-posedness for a one-dimensional Vlasov–Dirac–Benney equation with analytic initial data, using a contraction mapping in analytic function spaces, and explores stationary states and perturbations.

Contribution

It introduces a novel analytic framework for local existence and uniqueness of solutions to a nonlinear Vlasov equation with a real-analytic nonlinearity.

Findings

Established local-in-time existence and uniqueness for small analytic initial data.

Provided an energy-based characterization of stationary states.

Developed quantitative composition estimates for analytic nonlinearities.

Abstract

We study a one-dimensional nonlinear Vlasov equation with a local self-consistent force field generated by the density, where the force is given by the spatial derivative of a real-analytic nonlinearity. For small analytic initial data, we prove local-in-time existence and uniqueness of analytic solutions. In particular, this yields a perturbative well-posedness result around the trivial equilibrium. We also give an energy-based representation of weak stationary states and discuss perturbations around spatially homogeneous stationary profiles. The proof relies on a contraction mapping argument in a complete metric space of analytic functions. As a technical byproduct, we establish quantitative composition estimates for analytic nonlinearities in the analytic norms used in the argument.