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 κ > 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 κ > 0 . [ABSTRACT FROM AUTHOR]