October  2016, 21(8): 2767-2784. doi: 10.3934/dcdsb.2016072

On the Rayleigh-Taylor instability for the compressible non-isentropic inviscid fluids with a free interface

1. 

Department of Mathematics, Shanghai Normal University, Shanghai 200234

2. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240

Received  November 2014 Revised  June 2016 Published  September 2016

In this paper, we study the Rayleigh-Taylor instability phenomena for two compressible, immiscible, inviscid, ideal polytropic fluids. Such two kind of fluids always evolve together with a free interface due to the uniform gravitation. We construct the steady-state solutions for the denser fluid lying above the light one. With an assumption on the steady-state temperature function, we find some growing solutions to the related linearized problem, which in turn demonstrates the linearized problem is ill-posed in the sense of Hadamard. By such an ill-posedness result, we can finally prove the solutions to the original nonlinear problem does not have the property EE(k). Precisely, the $H^3$ solutions to the original nonlinear problem can not Lipschitz continuously depend on their initial data.
Citation: Jing Wang, Feng Xie. On the Rayleigh-Taylor instability for the compressible non-isentropic inviscid fluids with a free interface. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2767-2784. doi: 10.3934/dcdsb.2016072
References:
[1]

R. Duan, F. Jiang and S. Jiang, On the Rayleigh-Taylor instability for incompressible, inviscid magnetohydrodynamic flows, SIAM J. Appl. Math., 71 (2011), 1990-2013. doi: 10.1137/110830113.

[2]

Y. Guo and I. Tice, Linear Rayleigh-Taylor instability for viscous, compressible fluids, SIAM J. Math. Anal., 42 (2010), 1688-1720.

[3]

Y. Guo and I. Tice, Compressible, inviscid Rayleigh-Taylor instability, Indiana Univ. Math. J, 60 (2011), 677-711. doi: 10.1512/iumj.2011.60.4193.

[4]

H. Hwang and Y. Guo, On the dynamical Rayleigh-Taylor instability, Arch. Ration. Mech. Anal., 167 (2003), 235-253. doi: 10.1007/s00205-003-0243-z.

[5]

F. Jiang and S. Jiang, On linear instability and stability of the Rayleigh-Taylor problem in magnetohydrodynamics, J. Math. Fluid Mech., 17 (2015), 639-668. doi: 10.1007/s00021-015-0221-x.

[6]

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

[7]

J. Jang, I. Tice and Y. Wang, The compressible viscous surface-internal wave problem: Nonlinear Rayleigh-Taylor instability, Arch. Ration. Mech. Anal., 221 (2016), 215-272, arXiv:1501.07583. doi: 10.1007/s00205-015-0960-0.

[8]

H. Kull, Theory of the Rayleigh-Taylor instability, Phys. Rep., 206 (1991), 197-325. doi: 10.1016/0370-1573(91)90153-D.

[9]

L. Rayleigh, Analytic solutions of the Rayleigh equations for linear density profiles, Proc. London Math. Soc., 14 (1883), 170-177.

[10]

L. Rayleigh, Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density, in Scientific Paper, Cambridge University Press, Cambridge, UK, II (1990), 200-207.

[11]

G. I. Taylor, The instability of liquid surface when accelerated in a direction perpendicular to their planes, Proc. Roy Soc. London Ser. A, 201 (1950), 192-196. doi: 10.1098/rspa.1950.0052.

[12]

Y. J. Wang and I. Tice, The viscous surface-internal wave problem: Nonlinear Rayleigh-Taylor instability, Comm. Partial Differential Equations, 37 (2012), 1967-2028. doi: 10.1080/03605302.2012.699498.

[13]

Y. J. Wang, Critical magnetic number in the magnetohydrodynamic Rayleigh-Taylor instability, J. Math. Phys., 53 (2012), 073701, 22 pp. doi: 10.1063/1.4731479.

[14]

J. Wehausen and E. Laitone, Surface waves, Handbuch der Physik, Springer-Verlag, Berlin, 9 (1960), 446-778.

show all references

References:
[1]

R. Duan, F. Jiang and S. Jiang, On the Rayleigh-Taylor instability for incompressible, inviscid magnetohydrodynamic flows, SIAM J. Appl. Math., 71 (2011), 1990-2013. doi: 10.1137/110830113.

[2]

Y. Guo and I. Tice, Linear Rayleigh-Taylor instability for viscous, compressible fluids, SIAM J. Math. Anal., 42 (2010), 1688-1720.

[3]

Y. Guo and I. Tice, Compressible, inviscid Rayleigh-Taylor instability, Indiana Univ. Math. J, 60 (2011), 677-711. doi: 10.1512/iumj.2011.60.4193.

[4]

H. Hwang and Y. Guo, On the dynamical Rayleigh-Taylor instability, Arch. Ration. Mech. Anal., 167 (2003), 235-253. doi: 10.1007/s00205-003-0243-z.

[5]

F. Jiang and S. Jiang, On linear instability and stability of the Rayleigh-Taylor problem in magnetohydrodynamics, J. Math. Fluid Mech., 17 (2015), 639-668. doi: 10.1007/s00021-015-0221-x.

[6]

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

[7]

J. Jang, I. Tice and Y. Wang, The compressible viscous surface-internal wave problem: Nonlinear Rayleigh-Taylor instability, Arch. Ration. Mech. Anal., 221 (2016), 215-272, arXiv:1501.07583. doi: 10.1007/s00205-015-0960-0.

[8]

H. Kull, Theory of the Rayleigh-Taylor instability, Phys. Rep., 206 (1991), 197-325. doi: 10.1016/0370-1573(91)90153-D.

[9]

L. Rayleigh, Analytic solutions of the Rayleigh equations for linear density profiles, Proc. London Math. Soc., 14 (1883), 170-177.

[10]

L. Rayleigh, Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density, in Scientific Paper, Cambridge University Press, Cambridge, UK, II (1990), 200-207.

[11]

G. I. Taylor, The instability of liquid surface when accelerated in a direction perpendicular to their planes, Proc. Roy Soc. London Ser. A, 201 (1950), 192-196. doi: 10.1098/rspa.1950.0052.

[12]

Y. J. Wang and I. Tice, The viscous surface-internal wave problem: Nonlinear Rayleigh-Taylor instability, Comm. Partial Differential Equations, 37 (2012), 1967-2028. doi: 10.1080/03605302.2012.699498.

[13]

Y. J. Wang, Critical magnetic number in the magnetohydrodynamic Rayleigh-Taylor instability, J. Math. Phys., 53 (2012), 073701, 22 pp. doi: 10.1063/1.4731479.

[14]

J. Wehausen and E. Laitone, Surface waves, Handbuch der Physik, Springer-Verlag, Berlin, 9 (1960), 446-778.

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