Observation estimate for kinetic transport equations by diffusion approximation

Bardos, Claude; Phung, Kim Dang

Comptes Rendus. Mathématique, Tome 355 (2017) no. 6, pp. 640-664.

L'objet de cet article est l'observation (et aussi la continuation unique) pour des solutions d'équations cinétiques linéaires avec, comme opérateur de collision, soit un modèle simplifié de l'équation de la neutronique, soit un opérateur de Fokker–Planck linéarisé. À l'aide de l'approximation de la diffusion, une inégalité d'observation en un temps donné est obtenue. Elle dépend du libre parcours moyen (ou de l'opacité du milieu) et de la fréquence de la moyenne de la donnée initiale. En plus de l'approximation de la diffusion, on utilise l'inégalité d'observation en temps fixé pour la diffusion. Pour cette dernière, on propose une nouvelle démonstration directe avec des estimations à poids utilisant la paramétrix à l'ordre zéro du noyau de la chaleur.

We study the unique continuation property for the neutron transport equation and for a simplified model of the Fokker–Planck equation in a bounded domain with absorbing boundary condition. An observation estimate is derived. It depends on the smallness of the mean free path and the frequency of the velocity average of the initial data. The proof relies on the well-known diffusion approximation under convenience scaling and on the basic properties of this diffusion. Eventually, we propose a direct proof for the observation at one time of parabolic equations. It is based on the analysis of the heat kernel.

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