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

Ricardo J. Alonso (1) , Véronique Bagland (2) , Yingda Cheng (3) , Bertrand Lods (4)

1 Departamento de Matemàtica
2 LMBP - Laboratoire de Mathématiques Blaise Pascal
3 Department of Mathematics
4 Department of Statistics and Economics

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.

Mots clés: Boltzmann equation; self-similar solution; dynamical fixed point.; measure solutions

Domaines: Equations aux dérivées partielles [math.AP]

Ricardo J. Alonso, Véronique Bagland, Yingda Cheng, Bertrand Lods. One dimensional dissipative Boltzmann equation: measure solutions, cooling rate and self-similar profile. SIAM Journal on Mathematical Analysis, 2018, 50 (1), pp.1278-1321. ⟨10.1137/17M1136791⟩. ⟨hal-01155456⟩