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arXiv · 1505.07207

One dimensional dissipative Boltzmann equation: measure solutions, cooling rate and self-similar profile

Ricardo Alonso·Véronique Bagland·Yingda Cheng·Bertrand Lods
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Abstract

This manuscript investigates the following aspects of the one dimensional dissipative Boltzmann equation associated to variable hard-spheres kernel: (1) we show the optimal cooling rate of the model by a careful study of the system satisfied by the solution's moments, (2) give existence and uniqueness of measure solutions, and (3) prove the existence of a non-trivial self-similar profile, i.e. homogeneous cooling state, after appropriate scaling of the equation. The latter issue is based on compactness tools in the set of Borel measures. More specifically, we apply a dynamical fixed point theorem on a suitable stable set, for the model dynamics, of Borel measures.

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Ricardo Alonso, Véronique Bagland, Yingda Cheng, Bertrand Lods. 2015-05-27. One dimensional dissipative Boltzmann equation: measure solutions, cooling rate and self-similar profile. https://arxiv.org/abs/1505.07207

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