December  2021, 41(12): 5609-5632. 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 Published  December 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 and Continuous Dynamical Systems, 2021, 41 (12) : 5609-5632. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

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

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

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

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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