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Propagation of Uniform Upper Bounds for the Spatially Homogeneous Relativistic Boltzmann Equation

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Abstract

In this paper, we prove the propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation. These polynomial and exponential \(L^\infty \) bounds have been known to be a challenging open problem in relativistic kinetic theory. To accomplish this, we establish two types of estimates for the gain part of the collision operator. First, we prove a potential type estimate and a relativistic hyper-surface integral estimate. We then combine those estimates using the relativistic counterpart of the Carleman representation to derive uniform control of the gain term for the relativistic collision operator. This allows us to prove the desired propagation of the uniform bounds of the solution. We further present two applications of the propagation of the uniform upper bounds: another proof of the Boltzmann H-theorem, and the asymptotic convergence of solutions to the relativistic Maxwellian equilibrium.

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Acknowledgements

J. W. Jang was supported by the Korean IBS Project IBS-R003-D1. R. M. Strain was partially supported by the NSF Grants DMS-1764177 and DMS-1500916 of the USA. S.-B. Yun is supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1801-02.

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Jang, J.W., Strain, R.M. & Yun, SB. Propagation of Uniform Upper Bounds for the Spatially Homogeneous Relativistic Boltzmann Equation. Arch Rational Mech Anal 241, 149–186 (2021). https://doi.org/10.1007/s00205-021-01649-0

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