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

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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

Jing Wang ^1, and Feng Xie ^2,

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

Abstract
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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.

Keywords: Rayleigh-Taylor instability, ill-posedness., compressible inviscid fluids, non-isentropic, free boundary.

Mathematics Subject Classification: 35L04, 35L65, 76E1.

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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