This study investigates the global well-posedness of a coupled Navier–Stokes–Darcy model incorporating the Beavers–Joseph–Saffman–Jones interface boundary condition in two-dimensional Euclidean space. We establish the existence of global strong solutions for the system in both linear and nonlinear cases where porosity depends on pressure. When dealing with the time-dependent porous media, the primary challenge in obtaining closed prior estimates arises from the presence of complex, sharp interfaces. To address this issue, we employ the classical Trace Theorem. Such space-time variable coupled systems are crucial for understanding underground fluid flow.

This study focused on investigating the global well-posedness of a coupled Navier-Stokes-Darcy model with the Beavers-Joseph-Saffman-Jones interface boundary condition in the three-dimensional Euclidean space. By utilizing this approach, we successfully obtained the global strong solution of the system in the three-dimensional space. Furthermore, we demonstrated the exponential stability of this strong solution. The significance of such coupled systems lies in their pivotal role in the analysis of subsurface flow problems, particularly in the context of karst aquifers.