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 lambda when lambda goes to zero. A blow-up criterion involving the norm of the gradient of the velocity in L-1(0, T; L-infinity) 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 lambda goes to zero. The rate of the convergence in L-2 is of order lambda. Copyright (C) 2008 John Wiley & Sons, Ltd.