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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 |
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.
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. |
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. 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. |
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