We consider a coupled system of Keller-Segel type equations and the incompressible Navier-Stokes equations in spatial dimension two and three. In the previous work [19], we established the existence of a weak solution of a Fokker-Plank equation in the Wasserstein space using the optimal transportation technique. Exploiting this result, we constructed solutions of Keller-Segel-Navier-Stokes equations such that the density of biological organism belongs to the absolutely continuous curves in the Wasserstein space. In this work, we refine the result on the existence of a weak solution of a Fokker-Plank equation in the Wasserstein space. As a result, we construct solutions of Keller-Segel-Navier-Stokes equations under weaker assumptions on the initial data.
Categories: math.AP