Nonlinear instability for nonhomogeneous incompressible viscous fluids

作者：Jiang Fei^*; Jiang Song; Ni GuoXi

来源：Science China Mathematics, 2013, 56(4): 665-686.

DOI：10.1007/s11425-013-4587-z

• nonhomogeneous Navier-Stokes equations
• steady density profile
• Rayleigh-Taylor instability
• incompressible viscous flows

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 H (k) , 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.

• 出版日期2013-4
• 单位福州大学; 北京应用物理与计算数学研究所