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

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.