Inhomogeneous six-wave kinetic equation in exponentially weighted $L^\infty$ spaces

Nataša Pavlović; Maja Tasković; Luisa Velasco

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arXiv · 2501.10565

Inhomogeneous six-wave kinetic equation in exponentially weighted $L^\infty$ spaces

Nataša Pavlović·Maja Tasković·Luisa Velasco

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Abstract

Six-wave interactions are used for modeling various physical systems, including in optical wave turbulence [16] (where a cascade of photons displays this kind of behavior) and in quantum wave turbulence [31] (for the interaction of Kelvin waves in superfluids). In this paper, we initiate the analysis of the Cauchy problem for the spatially inhomogeneous six-wave kinetic equation. More precisely, we obtain the existence and uniqueness of non-negative mild solutions to this equation in exponentially weighted $L^\infty_{xv}$ spaces. This is accompanied by an analysis of the long-time behavior of such solutions - we prove that the solutions scatter, that is, they converge to solutions of the transport equation in the limit as $t \to \pm \infty$. Compared with the study of four-wave kinetic equations, the main challenge we face is to address the increased complexity of the geometry of the six-wave interactions.

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Nataša Pavlović, Maja Tasković, Luisa Velasco. 2025-01-17. Inhomogeneous six-wave kinetic equation in exponentially weighted $L^\infty$ spaces. https://arxiv.org/abs/2501.10565

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