Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion

In this paper, we study Cauchy problem of the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Taking advantage of a coupling structure of the equations and using the damping effect of the growth term g(n), we obtain the necessary estimates of solution (n, c, u) without the diffusion term ∆n. These uniform estimates enable us to establish the global-in-time existence of almost weak solutions for the system.