5 research outputs found
Nonlinear Instability for Nonhomogeneous Incompressible Viscous Fluids
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.Comment: 24 page