Rayleigh-Taylor Instability for Viscous Incompressible Capillary Fluids

We investigate the linear and nonlinear instability of a smooth Rayleigh-Taylor steady state solution to the three-dimensional incompressible Navier-Stokes-Korteweg equations in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady state solution and find that for any capillary coefficient kappa > 0, we can construct the solutions of the linearized problem that grow in time in Sobolev space H-m, thus leading to the linear instability. However, with the help of the constructed unstable solutions of the linearized problem, we just establish the nonlinear instability for small enough capillary coefficient kappa > 0.