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

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.

💡 Research Summary

The paper investigates the local-in-time analytic well‑posedness of a one‑dimensional nonlinear Vlasov equation with a self‑consistent force that is the spatial derivative of a real‑analytic function of the density. The model under study is
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