Local well-posedness in the Wasserstein space for a chemotaxis model coupled to incompressible fluid flows

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 [17], we established the existence of a weak solution of a Fokker–Planck 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–Planck equation in the Wasserstein space. As a result, we construct solutions of Keller–Segel–Navier–Stokes equations under weaker assumptions on the initial data.