偏微分方程式セミナー: The rigorous derivation of the wave kinetic equations for nonlinear dispersive equations, Kiyeon Lee 氏 (KAIST / University of Michigan) / TRICHOTOMY FOR MODIFIED SCATTERING ACROSS THE YUKAWA–COULOMB TRANSITION, Yonggeun Cho 氏 (Jeonbuk National University)

Time：15：30 – 18：00

Place：理学部 4 号館 4-501 (hybrid)

Organizer：喜多 航佑

Speaker1：Kiyeon Lee 氏 (KAIST / University of Michigan)

Title：The rigorous derivation of the wave kinetic equations for nonlinear dispersive equations

Abstract：The wave kinetic equation is expected to describe the long-time statistical behavior of weakly nonlinear dispersive waves and serves as a fundamental object in wave turbulence theory. In this talk, I will present an overview of the recent rigorous derivation of the wave kinetic equation for nonlinear Schrödinger equations due to Yu Deng and Zaher Hani (`19-`23). I will first introduce the basic kinetic picture and explain how diagrammatic expansions and resonance relations arise in the analysis. I will then discuss the molecule formalism, which provides an efficient framework for organizing the combinatorial structures appearing in the proof. Finally, I will briefly report on ongoing joint work concerning the derivation of the wave kinetic equation for the Klein–Gordon equation motivated by models of anharmonic crystals. This is joint work with Zaher Hani (University of Michigan) and Katja Vassilev (University of Chicago).

Speaker2：Yonggeun Cho氏 (Jeonbuk National University)

Title：TRICHOTOMY FOR MODIFIED SCATTERING ACROSS THE YUKAWA–COULOMB TRANSITION

Abstract：In this talk, we consider the long-time asymptotics of the three-dimensional Hartree equation with Yukawa potential
$$
V_\mu(x)=\frac{e^{-\mu |x|}}{|x|},
\qquad 0\leq\mu\leq1.
$$
The Coulomb case corresponds to \(\mu=0\), while \(\mu>0\) introduces the screening length \(\mu^{-1}\). We consider the singular two-parameter limit \(\mu \to 0\) and \(t \to \infty\). One may expect that the limit \(\mu \to 0\) simply recovers the usual Coulomb modified sacttering phase. But this is not the case. The asymptotic behavior depends on the comparison between the observation scale \(t\) and the screening length \(\mu^{-1}\), equivalently on the parameter \(\mu t\). This gives the three regimes \(\mu_n t_n\to0\), \(\mu_n t_n\to L\in(0,\infty)\), and \(\mu_n t_n\to\infty\).