Sharp decay thresholds in weighted L^∞ for wave kinetic equations with power-law dispersion

2026/07/16 by Xilu Zhu
#math.AP #math-ph #math.MP

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Abstract

We study four-wave kinetic equations in space dimension three with power-law dispersion ω(p)=|p|^a and collision kernels with high-frequency growth measured by β. In weighted L^∞ spaces, we identify the sharp decay threshold s_c=4β+3-\frac a2. For s>s_c, we prove local well-posedness by establishing trilinear bounds for the full gain-loss collision operator. For s<s_c, we prove ill-posedness by constructing data concentrated near a high-low-low-high resonant configuration. This threshold captures the balance between the high-frequency strength of the kernel and the geometry of the resonant manifold. The proof also shows that gain-loss cancellations are essential in the most delicate regimes.

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