Global existence and aggregation of chemotaxis–fluid systems in dimension two

Abstract

This paper investigates the coupled Patlak–Keller–Segel–Navier–Stokes (PKS–NS) system in two-dimensional bounded domains with a focus on global well-posedness and the formation of striking patterns, in which the boundary conditions are Neumann conditions for the cell density and the chemical concentration, and the Navier slip boundary condition with zero friction for the fluid velocity. We prove that the solution of the system exists globally in time in the case of subcritical mass. Concerning the critical mass case, we construct the boundary spot steady states rigorously via the inner-outer gluing method.While studying the global existence and the concentration phenomenon of the chemotaxis–fluid model, we develop the global W2,p theory for the 2D stationary Stokes system subject to Navier boundary conditions with zero friction and further establish semigroup estimates of the nonstationary counterpart by analyzing the Stokes eigenvalue problem.