Abstract: This paper investigates the blow-up criteria for the Navier-Stokes equations of inhomogeneous incompressible fluids. While the existence of weak solutions to these equations has been extensively studied, the existence of strong solutions remains an open question, particularly for highly regular data that meet specific compatibility conditions, where only local existence results are currently available. The focus of this study is on the blow-up criteria within a two-dimensional bounded smooth domain, which refers to the conditions that cause solutions to lose regularity. By deriving priori estimates independent of the density lower bounds and establishing the existence and uniqueness of local strong solutions to the initial value problem or the initial boundary value problem, this paper offers new theoretical insights into the stability and existence of solutions in the dynamics of inhomogeneous incompressible fluids.

Key words: nonhomogeneous, incompressible fluids, Navier-Stokes equation, blowup criterion