Nonlinear Instability for Nonhomogeneous Incompressible Viscous Fluids

TL;DR

This paper demonstrates the nonlinear instability of certain steady solutions in nonhomogeneous incompressible Navier-Stokes equations under gravity, highlighting the role of the third velocity component in the instability mechanism.

Contribution

It provides the first analysis showing that the third velocity component induces instability, extending understanding of Rayleigh-Taylor instability in viscous fluids.

Findings

Linearized solutions grow exponentially in Sobolev space

Nonlinear instability is established using unstable linear solutions

Third velocity component is crucial for instability

Abstract

We investigate the nonlinear instability of a smooth steady density profile solution of the threedimensional nonhomogeneous incompressible Navier-Stokes equations in the presence of a uniform gravitational field, including a Rayleigh-Taylor steady-state solution with heavier density with increasing height (referred to the Rayleigh-Taylor instability). We first analyze the equations obtained from linearization around the steady density profile solution. Then we construct solutions of the linearized problem that grow in time in the Sobolev space Hk, thus leading to a global instability result for the linearized problem. With the help of the constructed unstable solutions and an existence theorem of classical solutions to the original nonlinear equations, we can then demonstrate the instability of the nonlinear problem in some sense. Our analysis shows that the third component of the velocity already induces the instability, this is different from the previous known results.