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On the singular limit of a model transport semigroup
Author(s) -
MokhtarKharroubi M.,
Protopopescu V.,
Thevenot L.
Publication year - 2000
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/1099-1476(200010)23:15<1301::aid-mma166>3.0.co;2-6
Subject(s) - mathematics , semigroup , torus , limit (mathematics) , zero (linguistics) , mathematical analysis , functional calculus , diffusion equation , convection–diffusion equation , norm (philosophy) , scaling , scaling limit , operator (biology) , mathematical physics , pure mathematics , geometry , philosophy , linguistics , biochemistry , economy , chemistry , repressor , political science , transcription factor , law , economics , gene , service (business)
We consider the diffusion limit of a model transport equation on the torus or the whole space, as a scaling parameter ε (the mean free path), tends to zero. We show that, for arbitrary initial data $u_0(x,v)$\nopagenumbers\end , the solution converges in norm topology for each $t>0$\nopagenumbers\end , to the solution of a diffusion equation with initial data \def\d{{\rm d}}$u_D^0(x)=\int u_0(x,v)\,\d v$\nopagenumbers\end . The proof relies on Fourier analysis which diagonalizes the transport operator, a Dunford functional calculus and the analysis of the behaviour of the transport spectrum as ε tends to zero. Copyright © 2000 John Wiley & Sons, Ltd.