Instability and Stability Analysis of Three-Dimensional Nonhomogeneous Incompressible Viscous Flows with Navier-Slip Boundary Conditions

This article investigates the instability and stability of hydrostatic equilibria of the three-dimensional nonhomogeneous incompressible viscous flows subject to the Navier-slip boundary conditions in a bounded domain. First, we establish the linear instability of a steady-state solution with a density profile that increases in a specific region. Then, we prove the nonlinear instability of the hydrostatic equilibrium in the Hadamard sense by constructing a nonlinearly perturbed solution, achieved through a sequence of refined energy estimates. Finally, we also show the global linear and non-linear stability of the steady-state solution when the density profile decreases in the direction opposite to the gravitational force. Our findings provide important extensions to previous results on both (a) the two-dimensional case involving flat boundaries and constant slip coefficients for incompressible viscous flows, and (b) the more complex three-dimensional scenario with no-slip boundary conditions.