Viscous limits for one dimensional Fokker-Planck-Burgers type systems

We study here the solutions to simple kinetic modelling of one-dimensional two-phase flows. The gas obeys a conservation law with smooth E-1 flux and diffusion term. The dispersed phase obeys a linear transport-diffusion kine tic equation of Fokker-Planck type. The problem involves the kinematic visc osity of the disperse phase accounting for the turbulent dispersion of the particles and the usual one of the gas flow. The existence and uniqueness o f solutions to that problem is proved in the case where at least one of the two viscosity coefficients is positive. We prove existence of solutions to the limiting problem obtained when the two viscosity coefficients tend to zero. (C) 2001 Academie des sciences/Editions scientifiques et medicales El sevier SAS.