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Well‐posed problem of a pollutant model of the Kazhikhov–Smagulov type
Author(s) -
Fang Daoyuan,
Fang Lin
Publication year - 2009
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/mma.1093
Subject(s) - mathematics , uniqueness , type (biology) , norm (philosophy) , zero (linguistics) , diffusion , mathematical analysis , interval (graph theory) , euler's formula , reaction–diffusion system , rate of convergence , thermodynamics , combinatorics , physics , ecology , linguistics , philosophy , political science , law , biology , channel (broadcasting) , electrical engineering , engineering
In this paper, we consider a well‐posed problem of a pollutant model of the Kazhikhov–Smagulov type, which is derived by Bresch et al. ( J. Math. Fluid Mech. 2007; 9 :377–397). For proper smooth data, existence and uniqueness are stated on a time interval, which become independent of the diffusion coefficient λ when λ goes to zero. A blow‐up criterion involving the norm of the gradient of the velocity in L 1 (0, T ; L ∞ ) is also proved. Besides, we show that if the density‐dependent Euler system has a smooth solution on a given time interval [0, T 0 ] , then the pollutant model of the Kazhikhov–Smagulov type with the same data and small diffusion coefficient has a smooth solution on [0, T 0 ] . The diffusion solution tends to the Euler solution when the diffusion coefficient λ goes to zero. The rate of the convergence in L 2 is of order λ . Copyright © 2008 John Wiley & Sons, Ltd.