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

Nuno J. Alves, Peter Markowich, Athanasios E. Tzavaras

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.

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

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.

Paper Structure

This paper contains 21 sections, 22 theorems, 274 equations.

Table of Contents

  1. Introduction
  2. Main results and strategy
  3. Relation to the literature and contributions
  4. Organization of the paper
  5. Generalized Vlasov equations
  6. Hamiltonian flow
  7. Monokinetic distributions and Euler systems
  8. Analytic functions and associated norms
  9. Analytic functions and norms
  10. Algebra norm estimates
  11. Real-analytic nonlinearities
  12. Auxiliary analytic functions
  13. Local well-posedness: perturbations of the trivial solution
  14. Preliminary estimates
  15. A time-dependent estimate
  16. ...and 6 more sections

Key Result

Lemma 3.1

Let $f$ be analytic with radius of convergence $\Lambda$. Then, for each $0 < \lambda < \Lambda$, and, for $n \in \mathbb{N}_0$,

Theorems & Definitions (48)

  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Remark 3.5
  • Lemma 3.6
  • proof
  • Lemma 3.7
  • ...and 38 more