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Nonlinear stability and instability in the Rayleigh–Taylor problem of stratified compressible MHD fluids

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Abstract

We establish the stability/instability criteria for the stratified compressible magnetic Rayleigh–Taylor (RT) problem in Lagrangian coordinates. More precisely, under the stability condition \(\varXi <1\) (under such case, the third component of impressed magnetic field is not zero), we show the existence of a unique solution with an algebraic decay in time for the (compressible) magnetic RT problem with proper initial data. The stability result in particular shows that a sufficiently large (impressed) vertical magnetic field can inhibit the growth of the RT instability. On the other hand, if \(\varXi >1\), there exists an unstable solution to the magnetic RT problem in the Hadamard sense. This in particular shows that the RT instability still occurs when the strength of an magnetic field is small or the magnetic field is horizontal. Moreover, by analyzing the stability condition in the magnetic RT problem for vertical magnetic fields, we can observe that the compressibility destroys the stabilizing effect of magnetic fields in the vertical direction. Fortunately, the instability in the vertical direction can be inhibited by the stabilizing effect of the pressure, which also plays an important role in the proof of the stability of the magnetic RT problem. In addition, we extend the results for the magnetic RT problem to the compressible viscoelastic RT problem, and find that the stabilizing effect of elasticity is stronger than that of the magnetic fields.

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Acknowledgements

The authors would like to thank the anonymous referee for invaluable suggestions, which improve the presentation of this paper. The authors also thank Dr. Weicheng Zhang for pointing out Dirichlets approximation theorem.

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Authors and Affiliations

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

    Fei Jiang

  2. Key Laboratory of Operations Research and Control of Universities in Fujian, Fuzhou, 350108, China

    Fei Jiang

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

    Song Jiang

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Correspondence to Fei Jiang.

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Communicated by F. H. Lin.

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The research of Fei Jiang was supported by NSFC (Grant No. 11671086) and the NSF of Fujian Province of China (Grant No. 2016J06001), and the research of Song Jiang by NSFC (Grant Nos. 11631008 and 11371065).

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Jiang, F., Jiang, S. Nonlinear stability and instability in the Rayleigh–Taylor problem of stratified compressible MHD fluids. Calc. Var. 58, 29 (2019). https://doi.org/10.1007/s00526-018-1477-9

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