Journal Article

Nonlinear instability for nonhomogeneous incompressible viscous fluids

JIANG Fei JIANG Song NI GuoXi

2013

Overview

We investigate the nonlinear instability of a smooth steady density profile solution to the three- dimensional 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 to 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, which is different from the previous known results.

• Publisher: SP Science China Press,Springer Nature B.V
• Subject: Applications of Mathematics / Construction / Density / Existence theorems / Fluid flow / Gravitational fields / Incompressible flow / Instability / Linearization / Mathematical analysis / Mathematics / Mathematics and Statistics / Navier-Stokes equations / Nonlinear equations / Nonlinearity / Sobolev space / Sobolev空间 / Stability / Taylor instability / Viscous fluids / 不可压缩Navier-Stokes方程 / 不可压缩粘性流体 / 不稳定性 / 平稳密度 / 非均质 / 非线性不稳定 / 非线性方程