doi: 10.3934/dcds.2021090

Well-posedness of the two-phase flow problem in incompressible MHD

1. 

School of Mathematical Sciences, Peking University, Beijing 100871, China

2. 

Department of Mathematics, Zhejiang University, Hangzhou 310027, China

* Corresponding author: Hui Li

Received  September 2020 Revised  April 2021 Early access  June 2021

In this paper, we study the two phase flow problem in the ideal incompressible magnetohydrodynamics. We propose a Syrovatskij type stability condition, and prove the local well-posedness of the two phase flow problem with initial data satisfies such condition. This result shows that the magnetic field has a stabilizing effect on Kelvin-Helmholtz instability even the fluids on each side of the free interface have different densities.

Citation: Changyan Li, Hui Li. Well-posedness of the two-phase flow problem in incompressible MHD. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021090
References:
[1]

W. I. Axford, Note on a problem of magnetohydrodynamic stability, Canad. J. Phys., 40 (1962), 654-655.  doi: 10.1139/p62-064.  Google Scholar

[2]

T. AlazardN. Burq and C. Zuily, On the Cauchy problem for gravity water waves, Invent. Math., 198 (2014), 71-163.  doi: 10.1007/s00222-014-0498-z.  Google Scholar

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D. M. Ambrose and N. Masmoudi, Well-posedness of 3D vortex sheets with surface tension, Comm. Math. Sci., 5 (2007), 391-430.  doi: 10.4310/CMS.2007.v5.n2.a9.  Google Scholar

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C. ChengD. Coutand and S. Shkoller, Navier-Stokes equations interacting with a nonlinear elastic biofluid shell, SIAM J. Math. Anal., 39 (2007), 742-800.  doi: 10.1137/060656085.  Google Scholar

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V. CarboneG. Einaudi and P. Veltri, Effects of turbulence development in solar surges, Solar. Phys., 111 (1987), 31-44.  doi: 10.1007/978-94-009-3999-8_4.  Google Scholar

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D. Christodoulou and H. Lindblad, On the motion of the free surface of a liquid, Comm. Pure Appl. Math., 53 (2000), 1536-1602.  doi: 10.1002/1097-0312(200012)53:12<1536::AID-CPA2>3.0.CO;2-Q.  Google Scholar

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J.-F. CoulombelA. MorandoP. Secchi and P. Trebeschi, A priori estimates for 3D incompressible current-vortex sheets, Comm. Math. Phys., 311 (2012), 247-275.  doi: 10.1007/s00220-011-1340-8.  Google Scholar

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S. Friedlander and D. Serre, Handbook of Mathematical Fluid Dynamics, North Holland/Elsevier, 2004.  Google Scholar

[9]

F. JiangS. Jiang and Y. Wang, On the Rayleigh-Taylor instability for the incompressible viscous magnetohydrodynamic equations, Comm. Partial Differential Equations, 39 (2014), 399-438.  doi: 10.1080/03605302.2013.863913.  Google Scholar

[10]

A. L. La Belle-HamerZ. F. Fu and L. C. Lee, A mechanism for patchy reconnection at the dayside magnetopause, Geophysical Research Letters, 15 (1988), 152-155.  doi: 10.1029/GL015i002p00152.  Google Scholar

[11]

H. LiW. Wang and Z. Zhang, Well-posedness of the free boundary problem in incompressible elastodynamics, J. Differential Equations, 267 (2019), 6604-6643.  doi: 10.1016/j.jde.2019.07.001.  Google Scholar

[12] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27, Cambridge University Press, Cambridge, 2002.   Google Scholar
[13]

G. V. Miloshevsky and A. Hassanein, Modelling of Kelvin-Helmholtz instability and splashing of melt layers from plasma-facing components in tokamaks under plasma impact, Nuclear Fusion, 50 (2010), 115005. doi: 10.1088/0029-5515/50/11/115005.  Google Scholar

[14]

A. MorandoY. Trakhinin and P. Trebeschi, Stability of incompressible current-vortex sheets, J. Math. Anal. Appl., 347 (2008), 502-520.  doi: 10.1016/j.jmaa.2008.06.002.  Google Scholar

[15]

L. Ofman, X. L. Chen, P. J. Morrison and et al., Resistive tearing mode instability with shear flow and viscosity, Physics of Fluids B: Plasma Physics, 3 (1991), 1364. Google Scholar

[16]

S. I. Syrovatskij, The stability of tangential discontinuities in a magnetohydrodynamic medium, Z. Eksperim. Teoret. Fiz., 24 (1953), 622-629.   Google Scholar

[17]

Y. SunW. Wang and Z. Zhang, Nonlinear stability of the current-vortex sheet to the incompressible MHD equations, Comm. Pure Appl. Math., 71 (2018), 356-403.  doi: 10.1002/cpa.21710.  Google Scholar

[18]

J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler's equation, Comm. Pure Appl. Math., 61 (2008), 698-744.  doi: 10.1002/cpa.20213.  Google Scholar

[19]

J. Shatah and C. Zeng, A priori estimates for fluid interface problems, Comm. Pure Appl. Math., 61 (2008), 848-876.  doi: 10.1002/cpa.20241.  Google Scholar

[20]

Y. Trakhinin, On the existence of incompressible current-vortex sheets: study of a linearized free boundary value problem, Math. Methods Appl. Sci., 28 (2005), 917-945.  doi: 10.1002/mma.600.  Google Scholar

[21]

S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D, Invent. Math., 130 (1997), 39-72.  doi: 10.1007/s002220050177.  Google Scholar

[22]

S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445-495.  doi: 10.1090/S0894-0347-99-00290-8.  Google Scholar

[23]

H. Yang, Z. Xu, E. K. Lim, et al., Observation of the Kelvin-Helmholtz Instability in a Solar Prominence, The Astrophysical Journal, 857 (2018), 115. doi: 10.3847/1538-4357/aab789.  Google Scholar

[24]

P. Zhang and Z. Zhang, On the free boundary problem of three-dimensional incompressible Euler equations, Comm. Pure Appl. Math., 61 (2008), 877-940.  doi: 10.1002/cpa.20226.  Google Scholar

show all references

References:
[1]

W. I. Axford, Note on a problem of magnetohydrodynamic stability, Canad. J. Phys., 40 (1962), 654-655.  doi: 10.1139/p62-064.  Google Scholar

[2]

T. AlazardN. Burq and C. Zuily, On the Cauchy problem for gravity water waves, Invent. Math., 198 (2014), 71-163.  doi: 10.1007/s00222-014-0498-z.  Google Scholar

[3]

D. M. Ambrose and N. Masmoudi, Well-posedness of 3D vortex sheets with surface tension, Comm. Math. Sci., 5 (2007), 391-430.  doi: 10.4310/CMS.2007.v5.n2.a9.  Google Scholar

[4]

C. ChengD. Coutand and S. Shkoller, Navier-Stokes equations interacting with a nonlinear elastic biofluid shell, SIAM J. Math. Anal., 39 (2007), 742-800.  doi: 10.1137/060656085.  Google Scholar

[5]

V. CarboneG. Einaudi and P. Veltri, Effects of turbulence development in solar surges, Solar. Phys., 111 (1987), 31-44.  doi: 10.1007/978-94-009-3999-8_4.  Google Scholar

[6]

D. Christodoulou and H. Lindblad, On the motion of the free surface of a liquid, Comm. Pure Appl. Math., 53 (2000), 1536-1602.  doi: 10.1002/1097-0312(200012)53:12<1536::AID-CPA2>3.0.CO;2-Q.  Google Scholar

[7]

J.-F. CoulombelA. MorandoP. Secchi and P. Trebeschi, A priori estimates for 3D incompressible current-vortex sheets, Comm. Math. Phys., 311 (2012), 247-275.  doi: 10.1007/s00220-011-1340-8.  Google Scholar

[8]

S. Friedlander and D. Serre, Handbook of Mathematical Fluid Dynamics, North Holland/Elsevier, 2004.  Google Scholar

[9]

F. JiangS. Jiang and Y. Wang, On the Rayleigh-Taylor instability for the incompressible viscous magnetohydrodynamic equations, Comm. Partial Differential Equations, 39 (2014), 399-438.  doi: 10.1080/03605302.2013.863913.  Google Scholar

[10]

A. L. La Belle-HamerZ. F. Fu and L. C. Lee, A mechanism for patchy reconnection at the dayside magnetopause, Geophysical Research Letters, 15 (1988), 152-155.  doi: 10.1029/GL015i002p00152.  Google Scholar

[11]

H. LiW. Wang and Z. Zhang, Well-posedness of the free boundary problem in incompressible elastodynamics, J. Differential Equations, 267 (2019), 6604-6643.  doi: 10.1016/j.jde.2019.07.001.  Google Scholar

[12] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27, Cambridge University Press, Cambridge, 2002.   Google Scholar
[13]

G. V. Miloshevsky and A. Hassanein, Modelling of Kelvin-Helmholtz instability and splashing of melt layers from plasma-facing components in tokamaks under plasma impact, Nuclear Fusion, 50 (2010), 115005. doi: 10.1088/0029-5515/50/11/115005.  Google Scholar

[14]

A. MorandoY. Trakhinin and P. Trebeschi, Stability of incompressible current-vortex sheets, J. Math. Anal. Appl., 347 (2008), 502-520.  doi: 10.1016/j.jmaa.2008.06.002.  Google Scholar

[15]

L. Ofman, X. L. Chen, P. J. Morrison and et al., Resistive tearing mode instability with shear flow and viscosity, Physics of Fluids B: Plasma Physics, 3 (1991), 1364. Google Scholar

[16]

S. I. Syrovatskij, The stability of tangential discontinuities in a magnetohydrodynamic medium, Z. Eksperim. Teoret. Fiz., 24 (1953), 622-629.   Google Scholar

[17]

Y. SunW. Wang and Z. Zhang, Nonlinear stability of the current-vortex sheet to the incompressible MHD equations, Comm. Pure Appl. Math., 71 (2018), 356-403.  doi: 10.1002/cpa.21710.  Google Scholar

[18]

J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler's equation, Comm. Pure Appl. Math., 61 (2008), 698-744.  doi: 10.1002/cpa.20213.  Google Scholar

[19]

J. Shatah and C. Zeng, A priori estimates for fluid interface problems, Comm. Pure Appl. Math., 61 (2008), 848-876.  doi: 10.1002/cpa.20241.  Google Scholar

[20]

Y. Trakhinin, On the existence of incompressible current-vortex sheets: study of a linearized free boundary value problem, Math. Methods Appl. Sci., 28 (2005), 917-945.  doi: 10.1002/mma.600.  Google Scholar

[21]

S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D, Invent. Math., 130 (1997), 39-72.  doi: 10.1007/s002220050177.  Google Scholar

[22]

S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445-495.  doi: 10.1090/S0894-0347-99-00290-8.  Google Scholar

[23]

H. Yang, Z. Xu, E. K. Lim, et al., Observation of the Kelvin-Helmholtz Instability in a Solar Prominence, The Astrophysical Journal, 857 (2018), 115. doi: 10.3847/1538-4357/aab789.  Google Scholar

[24]

P. Zhang and Z. Zhang, On the free boundary problem of three-dimensional incompressible Euler equations, Comm. Pure Appl. Math., 61 (2008), 877-940.  doi: 10.1002/cpa.20226.  Google Scholar

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