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arXiv · 2411.12868

On the ill-posedness of kinetic wave equations

Ioakeim Ampatzoglou·Tristan Léger
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Abstract

In this article we identify a sharp ill-posedness/well-posedness threshold for kinetic wave equations (KWE) derived from quasilinear Schr\"{o}dinger models. We show well-posedness using a collisional averaging estimate proved in our earlier work \cite{AmLe}. Ill-posedness manifests as instantaneous loss of smoothness for well-chosen initial data. We also prove that both the gain-only and full equation share the same well-posedness threhold, thus legitimizing a gain-only approach to solving 4-wave kinetic equations.

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Ioakeim Ampatzoglou, Tristan Léger. 2024-11-19. On the ill-posedness of kinetic wave equations. https://arxiv.org/abs/2411.12868

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