In this paper, we are concerned with the local-in-time well-posedness of a fluid-kinetic model in which the BGK model with density dependent collision frequency is coupled with the inhomogeneous Navier-Stokes equation through drag forces. To the best knowledge of authors, this is the first result on the existence of local-in-time smooth solution for particle-fluid model with nonlinear inter-particle operator for which the existence of time can be prolonged as the size of initial data gets smaller.
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