Smoothing effect of the homogeneous Boltzmann equation with measure valued initial datum
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 2, p. 429-442
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We justify the smoothing effect for measure valued solutions to the space homogeneous Boltzmann equation of Maxwellian type cross sections. This is the first rigorous proof of the smoothing effect for any measure valued initial data except the single Dirac mass, which gives the optimal description on the regularity of solutions for positive time, caused by the singularity in the cross section. The main new ingredient in the proof is the introduction of a time degenerate coercivity estimate by using the microlocal analysis.

• We discuss the Boltzmann equation with the angular singularity in the cross section. • The spatially homogeneous Cauchy problem is considered in the Maxwellian case. • The smoothing effect occurs for measures initial data except for a single Dirac mass. • The coercivity of the collision operator is derived from the angular singularity. • The time degenerate coercivity is available for the initial datum, two Dirac masses.

Nous justifions l'effet régularisant pour les solutions à valeurs mesures de l'équation de Boltzmann spatialement homogène dans le cas des molécules maxwelliennes. Il s'agit de la première preuve rigoureuse de l'effet régularisant pour toutes données initiales à valeurs mesures sauf la masse de Dirac seule, ce qui donne la description optimale de la regularité des solutions en temps positif à causée par la singularité dans le noyau de collision. Le principal ingrédient nouveau dans la preuve est l'introduction d'unc inégalité de coercivité dégénérée par rapport au temps en utilisant l'analyse microlocale.

DOI : https://doi.org/10.1016/j.anihpc.2013.12.004
Classification:  35Q20,  76P05,  35H20,  82B40,  82C40
Keywords: Boltzmann equation, Smoothing effect, Measure initial datum, Coercivity estimate
@article{AIHPC_2015__32_2_429_0,
     author = {Morimoto, Yoshinori and Yang, Tong},
     title = {Smoothing effect of the homogeneous Boltzmann equation with measure valued initial datum},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {32},
     number = {2},
     year = {2015},
     pages = {429-442},
     doi = {10.1016/j.anihpc.2013.12.004},
     zbl = {1321.35125},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2015__32_2_429_0}
}
Morimoto, Yoshinori; Yang, Tong. Smoothing effect of the homogeneous Boltzmann equation with measure valued initial datum. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 2, pp. 429-442. doi : 10.1016/j.anihpc.2013.12.004. http://www.numdam.org/item/AIHPC_2015__32_2_429_0/

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