Well-posed problem of a pollutant model of the Kazhikhov–Smagulov type

✍ Scribed by Daoyuan Fang; Lin Fang

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 242 KB
• Volume: 32
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1093

✦ Synopsis

Abstract

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.