[Inégalité d'observation pour des équations cinétiques linéaires par l'approximation de diffusion]
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.
Accepté le :
Publié le :
Claude Bardos 1 ; Kim Dang Phung 2
@article{CRMATH_2017__355_6_640_0,
author = {Claude Bardos and Kim Dang Phung},
title = {Observation estimate for kinetic transport equations by diffusion approximation},
journal = {Comptes Rendus. Math\'ematique},
pages = {640--664},
publisher = {Elsevier},
volume = {355},
number = {6},
year = {2017},
doi = {10.1016/j.crma.2017.04.017},
language = {en},
}
TY - JOUR AU - Claude Bardos AU - Kim Dang Phung TI - Observation estimate for kinetic transport equations by diffusion approximation JO - Comptes Rendus. Mathématique PY - 2017 SP - 640 EP - 664 VL - 355 IS - 6 PB - Elsevier DO - 10.1016/j.crma.2017.04.017 LA - en ID - CRMATH_2017__355_6_640_0 ER -
Claude Bardos; Kim Dang Phung. Observation estimate for kinetic transport equations by diffusion approximation. Comptes Rendus. Mathématique, Volume 355 (2017) no. 6, pp. 640-664. doi : 10.1016/j.crma.2017.04.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.04.017/
[1] Observability inequalities and measurable sets, J. Eur. Math. Soc., Volume 16 (2014) no. 11, pp. 2433-2475
[2] Diffusion approximation of radiative transfer problems with interfaces, SIAM J. Appl. Math., Volume 60 (2000) no. 6, pp. 1887-1912
[3] Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels ; théorèmes d'approximation ; applications à l'équations de tranport, Ann. Sci. Éc. Norm. Supér., Volume 3 (1970), pp. 185-233
[4] The diffusion approximation for the linear Boltzmann equation with vanishing scattering coefficient, Commun. Math. Sci., Volume 13 (2015) no. 3, pp. 641-671
[5] The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximation, J. Funct. Anal., Volume 77 (1988) no. 2, pp. 434-460
[6] Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc., Volume 284 (1984), pp. 617-649
[7] Sur l'unicité retrograde des équations paraboliques et quelques questions voisines, Arch. Ration. Mech. Anal., Volume 50 (1973), pp. 10-25
[8] Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Masson, Paris, 1985
[9] Backward uniqueness for parabolic operators with non-Lipschitz coefficients, Osaka J. Math., Volume 52 (2015) no. 3, pp. 793-815
[10] Doubling properties of caloric functions, Appl. Anal., Volume 85 (2006), pp. 205-223
[11] Global Carleman inequalities for parabolic systems and application to controllability, SIAM J. Control Optim., Volume 45 (2006) no. 4, pp. 1395-1446
[12] Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 17 (2000), pp. 583-616
[13] Controllability of Evolution Equations, Lecture Notes Series, vol. 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996
[14] Solutions globales de l'équation des ondes semi-linéaire critique à coefficients variables, Bull. Soc. Math. Fr., Volume 131 (2003) no. 1, pp. 1-22
[15] Inverse Problems for Partial Differential Equations, Springer, New York, 2006
[16] Asymptotic solution of neutron transport problems for small mean free paths, J. Math. Phys., Volume 15 (1974), pp. 75-81
[17] Contrôle exact de l'équation de la chaleur, Commun. Partial Differ. Equ., Volume 20 (1995) no. 1–2, pp. 335-356
[18] On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., Volume 18 (2012), pp. 712-747
[19] Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations, Arch. Ration. Mech. Anal., Volume 195 (2010) no. 3, pp. 953-990
[20] Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces, Invent. Math., Volume 183 (2011) no. 2, pp. 245-336
[21] Sur l'unicité rétrograde dans les problèmes mixtes paraboliques, Math. Scand., Volume 8 (1960), pp. 277-286
[22] Improperly Posed Problems in Partial Differential Equations, Regional Conference Series in Applied Mathematics, vol. 22, SIAM, 1975
[23] Quantitative unique continuation for the semilinear heat equation in a convex domain, J. Funct. Anal., Volume 259 (2010) no. 5, pp. 1230-1247
[24] An observability estimate for parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc., Volume 15 (2013) no. 2, pp. 681-703
[25] Unique continuation properties and quantitative estimates of unique continuation for parabolic equations, Handbook of Differential Equations: Evolutionary Equations, vol. 5, Elsevier/North-Holland, Amsterdam, 2009, pp. 421-500
[26] Unique continuation at the boundary for elliptic operators in Dini domains, Southeast Asian Bull. Math., Volume 26 (2002) no. 3, pp. 523-534
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