Some results on free boundary problems of incompressible ideal magnetohydrodynamics equations

Chengchun Hao; Tao Luo; Chengchun Hao; Tao Luo

doi:10.3934/era.2022021

Electronic Research Archive

2022, Volume 30, Issue 2: 404-424. doi: 10.3934/era.2022021

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Some results on free boundary problems of incompressible ideal magnetohydrodynamics equations

Chengchun Hao ^1,3,
Tao Luo ^{2
,
,}

1.
HLM, Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
2.
Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
3.
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received: 31 August 2021 Revised: 01 November 2021 Accepted: 06 November 2021 Published: 18 January 2022

We survey some recent results related to free boundary problems of incompressible ideal magnetohydrodynamics equations, and present the main ideas in the proofs of the ill-posedness in 2D when the Taylor sign condition is violated given ^[1], and the well-posedness of a linearized problem given in ^[2] in general $ n $-dimensions ($ n \geqslant 2 $) when the Taylor sign condition is satisfied and the free boundaries are closed.
- free boundary problems,
- incompressible ideal magnetohydrodynamics equations,
- ill-posedness,
- linearized problem,
- well-posedness
Citation: Chengchun Hao, Tao Luo. Some results on free boundary problems of incompressible ideal magnetohydrodynamics equations[J]. Electronic Research Archive, 2022, 30(2): 404-424. doi: 10.3934/era.2022021

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Abstract

We survey some recent results related to free boundary problems of incompressible ideal magnetohydrodynamics equations, and present the main ideas in the proofs of the ill-posedness in 2D when the Taylor sign condition is violated given ^[1], and the well-posedness of a linearized problem given in ^[2] in general $ n $-dimensions ($ n \geqslant 2 $) when the Taylor sign condition is satisfied and the free boundaries are closed.

References

[1]	C. Hao, T. Luo, Ill-posedness of free boundary problem of the incompressible ideal MHD, Commun. Math. Phys., 376 (2020), 259–286. https://doi.org/10.1007/s00220-019-03614-1 doi: 10.1007/s00220-019-03614-1
[2]	C. Hao, T. Luo, Well-posedness for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations, J. Differential Equations, 299 (2021), 542–601. https://doi.org/10.1016/j.jde.2021.07.030 doi: 10.1016/j.jde.2021.07.030
[3]	C. Hao, T. Luo, A priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 212 (2014), 805–847. https://doi.org/10.1007/s00205-013-0718-5 doi: 10.1007/s00205-013-0718-5
[4]	S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D, Invent. Math., 130 (1997), 39–72. https://doi.org/10.1007/s002220050177 doi: 10.1007/s002220050177
[5]	S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445–495. https://doi.org/10.1090/S0894-0347-99-00290-8 doi: 10.1090/S0894-0347-99-00290-8
[6]	T. Alazard, N. Burq, C. Zuily, On the water waves equations with surface tension, Duke Math. J., 158 (2011), 413–499. https://doi.org/10.1215/00127094-1345653 doi: 10.1215/00127094-1345653
[7]	T. Alazard, N. Burq, C. Zuily, On the Cauchy problem for gravity water waves, Invent. Math., 198 (2014), 71–163. https://doi.org/10.1007/s00222-014-0498-z doi: 10.1007/s00222-014-0498-z
[8]	D. M. Ambrose, N. Masmoudi, The zero surface tension limit of two-dimensional water waves, Comm. Pure Appl. Math., 58 (2005), 1287–1315. https://doi.org/10.1002/cpa.20085 doi: 10.1002/cpa.20085
[9]	K. Beyer, M. Günther, On the Cauchy problem for a capillary drop, I. Irrotational motion, Math. Methods Appl. Sci., 21 (1998), 1149–1183. https://doi.org/10.1002/(SICI)1099-1476(199808)21:12<1149::AID-MMA990>3.0.CO;2-C doi: 10.1002/(SICI)1099-1476(199808)21:12<1149::AID-MMA990>3.0.CO;2-C
[10]	H. Christianson, V. M. Hur, G. Staffilani, Strichartz estimates for the water-wave problem with surface tension, Comm. Partial Differential Equations, 35 (2010), 2195–2252. https://doi.org/10.1080/03605301003758351 doi: 10.1080/03605301003758351
[11]	D. Christodoulou, H. Lindblad, On the motion of the free surface of a liquid, Comm. Pure Appl. Math., 53 (2000), 1536–1602. https://doi.org/10.1002/1097-0312(200012)53:12<1536::AID-CPA2>3.0.CO;2-Q doi: 10.1002/1097-0312(200012)53:12<1536::AID-CPA2>3.0.CO;2-Q
[12]	D. Coutand, S. Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc., 20 (2007), 829–930. https://doi.org/10.1090/S0894-0347-07-00556-5 doi: 10.1090/S0894-0347-07-00556-5
[13]	D. G. Ebin, The equations of motion of a perfect fluid with free boundary are not well posed, Comm. Partial Differential Equations, 12 (1987), 1175–1201. https://doi.org/10.1080/03605308708820523 doi: 10.1080/03605308708820523
[14]	D. Lannes, The Water Waves Problem: Mathematical Analysis and Asympototics, Mathematical Surveys and Monographs, 188, American Mathematical Society, Providence, RI, 2013. https://doi.org/10.1090/surv/188
[15]	H. Lindblad, K. H. Nordgren, A priori estimates for the motion of a self-gravitating incompressible liquid with free surface boundary, J. Hyperbolic Differ. Equ., 6 (2009), 407–432. https://doi.org/10.1142/S021989160900185X doi: 10.1142/S021989160900185X
[16]	T. Poyferre, Q.-H. Nguyen, A paradifferential reduction for the gravity-capillary waves system at low regularity and applications, Bull. Soc. Math. France, 145 (2017), 643–710. https://doi.org/10.24033/bsmf.2750 doi: 10.24033/bsmf.2750
[17]	J. Shatah, C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation, Comm. Pure Appl. Math., 61 (2008), 698–744. https://doi.org/10.1002/cpa.20213 doi: 10.1002/cpa.20213
[18]	Y. Sun, W. Wang, Z. Zhang, Well-posedness of the plasma-vacuum interface problem for ideal incompressible MHD, Arch. Ration. Mech. Anal., 234 (2019), 81–113. https://doi.org/10.1007/s00205-019-01386-5 doi: 10.1007/s00205-019-01386-5
[19]	P. Zhang, Z. Zhang, On the free boundary problem of three-dimensional incompressible Euler equations, Comm. Pure Appl. Math., 61 (2008), 877–940. https://doi.org/10.1002/cpa.20226 doi: 10.1002/cpa.20226
[20]	Y. Trakhinin, Local existence for the free boundary problem for the non-relativistic and relativistic compressible Euler equations with a vacuum boundary condition, Commun. Pure Appl. Math., 62 (2009), 1551–1594. https://doi.org/10.1002/cpa.20282 doi: 10.1002/cpa.20282
[21]	H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. Math., 162 (2005), 109–194. https://doi.org/10.4007/annals.2005.162.109 doi: 10.4007/annals.2005.162.109
[22]	H. Lindblad, C. Luo, A priori estimates for the compressible Euler equations for a liquid with free surface boundary and the incompressible limit, Comm. Pure Appl. Math., 71 (2018), 1273–1333. https://doi.org/10.1002/cpa.21734 doi: 10.1002/cpa.21734
[23]	D. Coutand, J. Hole, S. Shkoller, Well-posedness of the free-boundary compressible 3-D Euler equations with surface tension and the zero surface tension limit, SIAM J. Math. Anal., 45 (2013), 3690–3767. https://doi.org/10.1137/120888697 doi: 10.1137/120888697
[24]	T. Luo, H. Zeng, On the free surface motion of highly subsonic heat-conducting inviscid flows, Arch. Ration. Mech. Anal., 240 (2021), 877–926. https://doi.org/10.1007/s00205-021-01624-9 doi: 10.1007/s00205-021-01624-9
[25]	S. H. Shapiro, S. A. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars, WILEY-VCH, 2004.
[26]	H. Zirin, Astrophysics of the Sun, Cambridge University Press, Cambridge, 1988.
[27]	J. P. Cox, R. T. Giuli, Principles of Stellar Structure, I., II., Gordon and Breach, New York, 1968.
[28]	C. Luo, J. Zhang, A regularity result for the incompressible magnetohydrodynamics equations with free surface boundary, Nonlinearity, 33 (2020), 1499–1527. https://doi.org/10.1088/1361-6544/ab60d9 doi: 10.1088/1361-6544/ab60d9
[29]	X. Gu, Y. Wang, On the construction of solutions to the free-surface incompressible ideal magnetohydrodynamic equations, J. Math. Pures Appl., 128 (2019), 1–41. https://doi.org/10.1016/j.matpur.2019.06.004 doi: 10.1016/j.matpur.2019.06.004
[30]	X. Gu, C. Luo, J. Zhang, Local well-posedness of the free-boundary incompressible magnetohydrodynamics with surface tension, preprint, arXiv: 2105.00596.
[31]	H. Lindblad, Well-posedness for the linearized motion of an incompressible liquid with free surface boundary, Comm. Pure Appl. Math., 56 (2003), 153–197. https://doi.org/10.1002/cpa.10055 doi: 10.1002/cpa.10055
[32]	D. Lee, Uniform estimate of viscous free-boundary magnetohydrodynamics with zero vacuum magnetic field, SIAM J. Math. Anal., 49 (2017), 2710–2789. https://doi.org/10.1137/16M1089794 doi: 10.1137/16M1089794
[33]	D. Lee, Initial value problem for the free-boundary magnetohydrodynamics with zero magnetic boundary condition, Commun. Math. Sci., 16 (2018), 589–615. https://doi.org/10.4310/CMS.2018.v16.n3.a1 doi: 10.4310/CMS.2018.v16.n3.a1
[34]	P. Chen, S. Ding, Inviscid limit for the free-boundary problems of MHD equations with or without surface tension, preprint, arXiv: 1905.13047.
[35]	Y. Trakhinin, T. Wang, Well-posedness of free boundary problem in non-relativistic and relativistic ideal compressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 239 (2021), 1131–1176. https://doi.org/10.1007/s00205-020-01592-6 doi: 10.1007/s00205-020-01592-6
[36]	Y. Trakhinin, T. Wang, Well-posedness of the free boundary problem in ideal compressible magnetohydrodynamics with surface tension, Math. Ann., (2021). https://doi.org/10.1007/s00208-021-02180-z
[37]	C. Hao, On the motion of free interface in ideal incompressible MHD, Arch. Ration. Mech. Anal., 224 (2017), 515–553. https://doi.org/10.1007/s00205-017-1082-7 doi: 10.1007/s00205-017-1082-7
[38]	Y. Trakhinin, The existence of current-vortex sheets in ideal compressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 191 (2009), 245–310. https://doi.org/10.1007/s00205-008-0124-6 doi: 10.1007/s00205-008-0124-6
[39]	P. Secchi, Y. Trakhinin, Well-posedness of the plasma-vacuum interface problem, Nonlinearity, 27 (2014), 105–169. https://doi.org/10.1088/0951-7715/27/1/105 doi: 10.1088/0951-7715/27/1/105
[40]	A. Morando, Y. Trakhinin, P. Trebeschi, Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible MHD, Quart. Appl. Math., 72 (2014), 549–587. https://doi.org/10.1090/S0033-569X-2014-01346-7 doi: 10.1090/S0033-569X-2014-01346-7
[41]	Y. Trakhinin, Existence of compressible current-vortex sheets: variable coefficients linear analysis, Arch. Ration. Mech. Anal., 177 (2005), 331–366. https://doi.org/10.1007/s00205-005-0364-7 doi: 10.1007/s00205-005-0364-7
[42]	G. Q. Chen, Y. G. Wang, Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics, Arch. Ration. Mech. Anal., 187 (2008), 369–408. https://doi.org/10.1007/s00205-007-0070-8 doi: 10.1007/s00205-007-0070-8
[43]	Y.-G. Wang, F. Yu, Stabilization effect of magnetic fields on two-dimensional compressible current-vortex sheets, Arch. Ration. Mech. Anal., 208 (2013), 341–389. https://doi.org/10.1007/s00205-012-0601-9 doi: 10.1007/s00205-012-0601-9
[44]	Y. Sun, W. Wang, Z. Zhang, Nonlinear stability of current-vortex sheet to the incompressible MHD equations, Commun. Pure Appl. Math., 71 (2018), 356–403. https://doi.org/10.1002/cpa.21710 doi: 10.1002/cpa.21710
[45]	Y. Wang, Z. Xin, Global well-posedness of free interface problems for the incompressible inviscid resistive MHD, preprint, arXiv: 2009.11636.

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