arXiv subjects

Mengyi Xie

Publications and source records attributed to Mengyi Xie.

3 recordsLinked to original sources

arXiv2026

Long-time dynamics of Vlasov--Hartree systems across the Coulomb threshold

We study the three-dimensional Vlasov--Hartree system with Coulomb and inverse-power interactions. For small, regular, localized data, we prove global well-posedness and determine long-time dynamics. The main result concerns the Coulomb case, where the coupling persists at leading order through reciprocal asymptotic corrections: the fermionic distribution scatters along logarithmically corrected free characteristics set by the asymptotic bosonic profile, while the bosonic wave gains a logarithmic phase set by the asymptotic fermionic density. We also treat longer- and shorter-range interactions. Our proof combines Hamiltonian methods for the Vlasov equation with purely physical-space vector-field methods for dispersive equations. For stronger long-range interactions, we introduce a symplectic transformation and a regularized effective-phase cancellation, which remove leading nonintegrable interactions while preserving Hamiltonian structure and avoiding derivative loss.

math.AP↗

arXiv2026

Inverse Modified Scattering for Cubic NLS with a Repulsive Delta Potential

We study the one-dimensional cubic nonlinear Schrödinger equation with a repulsive delta potential and a localized inhomogeneous coefficient. We prove small-data modified scattering and construct the associated vector-valued modified scattering map, whose two components encode the coupling of the frequencies $ξ$ and $-ξ$ induced by the point interaction. We show that this map determines both the strength of the delta potential and the inhomogeneous coefficient. Quantitatively, the delta strength is recovered with Lipschitz stability, while the inhomogeneous coefficients satisfy a Hölder stability estimate. To our knowledge, these are the first recovery and stability results for a modified scattering map in the presence of an external potential.

math.AP↗

arXiv2025

Global Solutions for 5D Quadratic Fourth-Order Schrödinger Equations

We prove small data scattering for the fourth-order Schrödinger equation with quadratic nonlinearity \begin{equation*} i\partial_t u+Δ^2 u+αu^2 + β\bar{u}^2=0\qquad\text{in }\mathbb{R}^5 \end{equation*} for $α, β\in \mathbb{R}$. We extend the space-time resonance method, originally introduced by Germain, Masmoudi, and Shatah, to the setting involving the bilaplacian. We show that under a smallness condition on the initial data measured in a suitable norm, the solution satisfies $\|u\|_{L^{\infty}_x }\lesssim t^{-\frac{5}{4}} $ and scatters to the solution to the free equation. Although our work builds upon an established method, the fourth-order nature of the equation presents substantial challenges, requiring different techniques to overcome them.

math.AP↗