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Rigorous Study of the Equilibria of Collision Kernels Appearing in the Theory of Weak Turbulence

Abstract

In this paper, we rigorously obtain all the equilibria of collision kernels of type “two particles give two particles” appearing in weak turbulence theory under very general assumptions, thus completing the “equality case” in Boltzmann’s H-theorem for those models. We also provide some rigorous results for collision kernels of type “two particles give one particle”, under assumptions which include some of the most classical kernels of this type. The method of proof is inspired by the quantitative estimates obtained for the Landau equation in (J Funct Anal 269(5): 1359–1403, 2015).

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Acknowledgements

The research leading to this paper was partly funded by Université Sorbonne Paris Cité, in the framework of the “Investissements d’Avenir”, convention ANR-11-IDEX-0005. MB also acknowledges partial support from a Lichtenberg Professorship grant of the VolkswagenStiftung awarded to C. Kuehn.

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Correspondence to M. Breden.

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Breden, M., Desvillettes, L. Rigorous Study of the Equilibria of Collision Kernels Appearing in the Theory of Weak Turbulence. Arch Rational Mech Anal 235, 1151–1176 (2020). https://doi.org/10.1007/s00205-019-01441-1

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