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Nonlinear instability for nonhomogeneous incompressible viscous fluids

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Abstract

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.

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Authors and Affiliations

  1. College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, China

    Fei Jiang

  2. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Fei Jiang, Song Jiang & GuoXi Ni

Authors
  1. Fei Jiang
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  2. Song Jiang
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  3. GuoXi Ni
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Correspondence to Fei Jiang.

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Jiang, F., Jiang, S. & Ni, G. Nonlinear instability for nonhomogeneous incompressible viscous fluids. Sci. China Math. 56, 665–686 (2013). https://doi.org/10.1007/s11425-013-4587-z

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  • DOI: https://doi.org/10.1007/s11425-013-4587-z

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