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

Changyan Li; Hui Li

doi:10.3934/dcds.2021090

Article Contents

2021, Volume 41, Issue 12: 5609-5632. Doi: 10.3934/dcds.2021090 open access

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

Changyan Li^1, and
Hui Li^2, ,

1.
School of Mathematical Sciences, Peking University, Beijing 100871, China
2.
Department of Mathematics, Zhejiang University, Hangzhou 310027, China

^* Corresponding author: Hui Li
^* Corresponding author: Hui Li

Received: August 31, 2020

Revised: March 31, 2021

Early access: June 2021

Published: December 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q35; Secondary: 76W05.

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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. Alazard, N. 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. Cheng, D. 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. Carbone, G. 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. Coulombel, A. Morando, P. 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. Jiang, S. 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-Hamer, Z. 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. Li, W. 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. Morando, Y. 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. Sun, W. 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.