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arXiv · 2405.03984

Inhomogeneous wave kinetic equation and its hierarchy in polynomially weighted $L^\infty$ spaces

Ioakeim Ampatzoglou·Joseph K. Miller·Nataša Pavlović·Maja Tasković
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Abstract

Inspired by ideas stemming from the analysis of the Boltzmann equation, in this paper we expand well-posedness theory of the spatially inhomogeneous 4-wave kinetic equation, and also analyze an infinite hierarchy of PDE associated with this nonlinear equation. More precisely, we show global in time well-posedness of the spatially inhomogeneous 4-wave kinetic equation for polynomially decaying initial data. For the associated infinite hierarchy, we construct global in time solutions using the solutions of the wave kinetic equation and the Hewitt-Savage theorem. Uniqueness of these solutions is proved by using a combinatorial board game argument tailored to this context, which allows us to control the factorial growth of the Dyson series.

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Ioakeim Ampatzoglou, Joseph K. Miller, Nataša Pavlović, Maja Tasković. 2024-05-07. Inhomogeneous wave kinetic equation and its hierarchy in polynomially weighted $L^\infty$ spaces. https://arxiv.org/abs/2405.03984

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