American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[1]

Lvqiao liu, Lan Zhang. Optimal decay to the non-isentropic compressible micropolar fluids. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4575-4598. doi: 10.3934/cpaa.2020207
[2]

Fei Jiang, Song Jiang, Weiwei Wang. Nonlinear Rayleigh-Taylor instability for nonhomogeneous incompressible viscous magnetohydrodynamic flows. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1853-1898. doi: 10.3934/dcdss.2016076
[3]

Ling-Bing He, Li Xu. On the compressible Navier-Stokes equations in the whole space: From non-isentropic flow to isentropic flow. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3489-3530. doi: 10.3934/dcds.2021005
[4]

Pooja Girotra, Jyoti Ahuja, Dinesh Verma. Analysis of Rayleigh Taylor instability in nanofluids with rotation. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021018
[5]

Hong Cai, Zhong Tan, Qiuju Xu. Time periodic solutions of the non-isentropic compressible fluid models of Korteweg type. Kinetic & Related Models, 2015, 8 (1) : 29-51. doi: 10.3934/krm.2015.8.29
[6]

Zhong Tan, Leilei Tong. Asymptotic behavior of the compressible non-isentropic Navier-Stokes-Maxwell system in $\mathbb{R}^3$. Kinetic & Related Models, 2018, 11 (1) : 191-213. doi: 10.3934/krm.2018010
[7]

Huancheng Yao, Haiyan Yin, Changjiang Zhu. Stability of rarefaction wave for the compressible non-isentropic Navier-Stokes-Maxwell equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1297-1317. doi: 10.3934/cpaa.2021021
[8]

Zhong Tan, Yong Wang, Xu Zhang. Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$. Kinetic & Related Models, 2012, 5 (3) : 615-638. doi: 10.3934/krm.2012.5.615
[9]

Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673
[10]

Jiang Xu, Wen-An Yong. Zero-relaxation limit of non-isentropic hydrodynamic models for semiconductors. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1319-1332. doi: 10.3934/dcds.2009.25.1319
[11]

Kunquan Li, Yaobin Ou. Global wellposedness of vacuum free boundary problem of isentropic compressible magnetohydrodynamic equations with axisymmetry. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021052
[12]

Van-Sang Ngo, Stefano Scrobogna. Dispersive effects of weakly compressible and fast rotating inviscid fluids. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 749-789. doi: 10.3934/dcds.2018033
[13]

Xiangdi Huang, Zhouping Xin. On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4477-4493. doi: 10.3934/dcds.2016.36.4477
[14]

Haibo Cui, Zhensheng Gao, Haiyan Yin, Peixing Zhang. Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: Existence, stability and convergence rate. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4839-4870. doi: 10.3934/dcds.2016009
[15]

Dominic Breit, Eduard Feireisl, Martina Hofmanová. Generalized solutions to models of inviscid fluids. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3831-3842. doi: 10.3934/dcdsb.2020079
[16]

Yaodan Huang, Zhengce Zhang, Bei Hu. Bifurcation from stability to instability for a free boundary tumor model with angiogenesis. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2473-2510. doi: 10.3934/dcds.2019105
[17]

Serena Dipierro, Enrico Valdinoci. (Non)local and (non)linear free boundary problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 465-476. doi: 10.3934/dcdss.2018025
[18]

Paolo Secchi. An alpha model for compressible fluids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 351-359. doi: 10.3934/dcdss.2010.3.351
[19]

Ping Chen, Daoyuan Fang, Ting Zhang. Free boundary problem for compressible flows with density--dependent viscosity coefficients. Communications on Pure & Applied Analysis, 2011, 10 (2) : 459-478. doi: 10.3934/cpaa.2011.10.459
[20]

George Avalos, Pelin G. Geredeli, Justin T. Webster. Semigroup well-posedness of a linearized, compressible fluid with an elastic boundary. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1267-1295. doi: 10.3934/dcdsb.2018151

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (80)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences