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Modified scattering for the Vlasov-Riesz system with long-range interactions

Younghun Hong, Stephen Pankavich

Abstract

We study the long-time asymptotic behavior of small-data solutions to the three-dimensional Vlasov--Riesz system with the inverse power-law potential $λ|x|^{-α}$ in the strictly long-range regime ($0 < α< 1$). By introducing finite- and infinite-time modified wave operators for the characteristic flows, we describe the asymptotic dynamics via convergence to an effective profile along a suitably modified reference flow, and establish modified scattering of solutions. Our proof relies mainly on ODE techniques for the characteristic flows, while also using PDE methods for weighted $W^{1,\infty}$-bounds. Compared with the earlier result (of Huang and Kwon), our Lagrangian approach extends modified scattering to the broader regime $\frac{1}{2}<α<1$ and provides a distinct and more robust argument.

Modified scattering for the Vlasov-Riesz system with long-range interactions

Abstract

We study the long-time asymptotic behavior of small-data solutions to the three-dimensional Vlasov--Riesz system with the inverse power-law potential in the strictly long-range regime (). By introducing finite- and infinite-time modified wave operators for the characteristic flows, we describe the asymptotic dynamics via convergence to an effective profile along a suitably modified reference flow, and establish modified scattering of solutions. Our proof relies mainly on ODE techniques for the characteristic flows, while also using PDE methods for weighted -bounds. Compared with the earlier result (of Huang and Kwon), our Lagrangian approach extends modified scattering to the broader regime and provides a distinct and more robust argument.

Paper Structure

This paper contains 14 sections, 12 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Model description
  3. Statement of the main result
  4. Organization of the paper
  5. Notation
  6. Acknowledgement
  7. Global well-posedness with dispersion bounds
  8. Proof of global well-posedness
  9. Bounds for the forward-in-time characteristic flows
  10. Construction of the modified wave operator for characteristic flows
  11. Construction of the modified reference flow
  12. Finite-time modified wave operator
  13. Weighted $W^{1,\infty}$-estimates for modified distributions
  14. Proof of modified scattering

Key Result

Theorem 1.1

For $\frac{1}{2} < \alpha < 1$, there exists a small constant $\eta_* > 0$ such that the following holds. Suppose that $f_0 \ge 0$ and and let $f(t) \in C([0,\infty); W^{1,1}_{x,v}(\mathbb{R}^6) \cap W^{1,\infty}_{x,v}(\mathbb{R}^6))$ be the global solution to the Vlasov--Riesz system eq: VR with initial data $f_0$ such that eq: dispersion estimates-0 holds. Then, there exists an asymptotic profi

Theorems & Definitions (33)

  • Theorem 1.1: Modified scattering
  • Remark 1.2: Modified scattering in the strictly long-range regime
  • Remark 1.3: Quantum mechanical analogy
  • Remark 1.4: Initial data condition
  • Theorem 2.1: Global well-posedness for small data solutions
  • Lemma 2.2: Dispersion bounds imply characteristic flow estimates
  • proof
  • Remark 2.3
  • Lemma 2.4: Characteristic flow bounds imply dispersion estimates
  • proof
  • ...and 23 more