March  2019, 18(2): 569-602. doi: 10.3934/cpaa.2019029

Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the non-collinearity condition

School of Mathematics, Shanghai University of Finance and Economics, Shanghai Center of Mathematical Sciences, China

Received  November 2017 Revised  June 2018 Published  October 2018

Fund Project: The author is supported by NSFC grant 11601305.

We consider a free boundary problem for the axially symmetric incompressible ideal magnetohydrodynamic equations that describe the motion of the plasma in vacuum. Both the plasma magnetic field and vacuum magnetic field are tangent along the plasma-vacuum interface. Moreover, the vacuum magnetic field is composed in a non-simply connected domain and hence is non-trivial. Under the non-collinearity condition for the plasma and vacuum magnetic fields, we prove the local well-posedness of the problem in Sobolev spaces.

Citation: Xumin Gu. Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the non-collinearity condition. Communications on Pure and Applied Analysis, 2019, 18 (2) : 569-602. doi: 10.3934/cpaa.2019029
References:
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T. Alazard and J. M. Delort, Global solutions and asymptotic behavior for two dimensional gravity water waves, Ann. Sci. Éc. Norm. Supér., 48 (2015), 1149-1238.  doi: 10.24033/asens.2268.

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S. Alinhac, Existence d'ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, (French. English summary) [Existence of rarefaction waves for multidimensional hyperbolic quasilinear systems] Comm. Partial Differential Equations, 14 (1989), 173–230. doi: 10.1080/03605308908820595.

[3]

G. Chen and Y. Wang, Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics, Arch. Ration. Mech. Anal., 187 (2008), 369-408.  doi: 10.1007/s00205-007-0070-8.

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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>3.3.CO;2-H.

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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.

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D. Coutand and S. Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc., 20 (2007), 829-930.  doi: 10.1090/S0894-0347-07-00556-5.

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D. Coutand and S. Shkoller, A simple proof of well-posedness for the free-surface incompressible Euler equations, Discrete Contin. Dyn. Syst. Ser. S., 3 (2010), 429-449.  doi: 10.3934/dcdss.2010.3.429.

[8]

X. Gu and Z. Lei, Well-posedness of 1-D compressible Euler-Poisson equations with physical vacuum, J. Differential Equations, 252 (2012), 2160-2188.  doi: 10.1016/j.jde.2011.10.019.

[9]

P. GermainN. Masmoudi and J. Shatah, Global solutions for the gravity water waves equation in dimension 3, Ann. of Math. (2), 175 (2012), 691-754.  doi: 10.4007/annals.2012.175.2.6.

[10]

P. GermainN. Masmoudi and J. Shatah, Global solutions for capillary waves equation, Comm. Pure Appl. Math., 68 (2015), 625-687.  doi: 10.1002/cpa.21535.

[11]

X. Gu and Y. Wang, On the construction of solutions to the free-surface incompressible ideal magnetohydrodynamic equations, preprint, arXiv: 1609.07013.

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J. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics with Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, Cambridge, 2004.

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C. Hao and T. Luo, A priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 212 (2014), 805-847.  doi: 10.1007/s00205-013-0718-5.

[14]

A. Ionescu and F. Pusateri, Global solutions for the gravity water waves system in 2D, Invent. Math., 199 (2015), 653-804.  doi: 10.1007/s00222-014-0521-4.

[15]

A. Ionescu and F. Pusateri, Global regularity for 2D water waves with surface tension, Mem. Amer. Math. Soc., to appear.

[16]

D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654.  doi: 10.1090/S0894-0347-05-00484-4.

[17]

H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2), 162 (2005), 109-194.  doi: 10.4007/annals.2005.162.109.

[18]

N. Masmoudi and F. Rousset, Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equations, Arch. Ration. Mech. Anal., 223 (2017), 301-417.  doi: 10.1007/s00205-016-1036-5.

[19]

A. MorandoY. Trakhinin and P. Trebeschi, Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible MHD, Quart. Appl. Math., 72 (2014), 549-587.  doi: 10.1090/S0033-569X-2014-01346-7.

[20]

V. I. Nalimov, The Cauchy-Poisson problem, (Russian) Dinamika Splošn. Sredy Vyp. 18 Dinamika Židkost. so Svobod. Granicami., 254 (1974), 104–210.

[21]

P. Secchi and Y. Trakhinin, Well-posedness of the linearized plasma-vacuum interface problem, Interfaces Free Bound., 15 (2013), 323-357.  doi: 10.4171/IFB/305.

[22]

P. Secchi and Y. Trakhinin, Well-posedness of the plasma-vacuum interface problem, Nonlinearity, 27 (2014), 105-169.  doi: 10.1088/0951-7715/27/1/105.

[23]

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

[24]

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

[25]

Y. Sun, W. Wang and Z. Zhang, Well-posedness of the plasma-vacuum interface problem for ideal incompressible MHD, preprint, arXiv: 1705.00418.

[26]

M. Taylor, Partial Differential Equations, Vol. I-III, Berlin-Heidelberg-New York, Springer, 1996.1996.

[27]

Y. Trakhinin, The existence of current-vortex sheets in ideal compressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 191 (2009), 245-310.  doi: 10.1007/s00205-008-0124-6.

[28]

Y. Trakhinin, On the well-posedness of a linearized plasma-vacuum interface problem in ideal compressible MHD, J. Differential Equations, 249 (2010), 2577-2599.  doi: 10.1016/j.jde.2010.06.007.

[29]

Y. J. Wang and Z. Xin, Vanishing viscosity and surface tension limits of incompressible viscous surface waves, preprint, arXiv: 1504.00152. doi: 10.1007/s00220-014-1986-0.

[30]

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.

[31]

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.

[32]

S. Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135.  doi: 10.1007/s00222-009-0176-8.

[33]

S. Wu, Global wellposedness of the 3-D full water wave problem, Invent. Math., 184 (2011), 125-220.  doi: 10.1007/s00222-010-0288-1.

[34]

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:
[2]

S. Alinhac, Existence d'ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, (French. English summary) [Existence of rarefaction waves for multidimensional hyperbolic quasilinear systems] Comm. Partial Differential Equations, 14 (1989), 173–230. doi: 10.1080/03605308908820595.

[3]

G. Chen and Y. Wang, Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics, Arch. Ration. Mech. Anal., 187 (2008), 369-408.  doi: 10.1007/s00205-007-0070-8.

[4]

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>3.3.CO;2-H.

[5]

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.

[6]

D. Coutand and S. Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc., 20 (2007), 829-930.  doi: 10.1090/S0894-0347-07-00556-5.

[7]

D. Coutand and S. Shkoller, A simple proof of well-posedness for the free-surface incompressible Euler equations, Discrete Contin. Dyn. Syst. Ser. S., 3 (2010), 429-449.  doi: 10.3934/dcdss.2010.3.429.

[8]

X. Gu and Z. Lei, Well-posedness of 1-D compressible Euler-Poisson equations with physical vacuum, J. Differential Equations, 252 (2012), 2160-2188.  doi: 10.1016/j.jde.2011.10.019.

[9]

P. GermainN. Masmoudi and J. Shatah, Global solutions for the gravity water waves equation in dimension 3, Ann. of Math. (2), 175 (2012), 691-754.  doi: 10.4007/annals.2012.175.2.6.

[10]

P. GermainN. Masmoudi and J. Shatah, Global solutions for capillary waves equation, Comm. Pure Appl. Math., 68 (2015), 625-687.  doi: 10.1002/cpa.21535.

[11]

X. Gu and Y. Wang, On the construction of solutions to the free-surface incompressible ideal magnetohydrodynamic equations, preprint, arXiv: 1609.07013.

[12]

J. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics with Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, Cambridge, 2004.

[13]

C. Hao and T. Luo, A priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 212 (2014), 805-847.  doi: 10.1007/s00205-013-0718-5.

[14]

A. Ionescu and F. Pusateri, Global solutions for the gravity water waves system in 2D, Invent. Math., 199 (2015), 653-804.  doi: 10.1007/s00222-014-0521-4.

[15]

A. Ionescu and F. Pusateri, Global regularity for 2D water waves with surface tension, Mem. Amer. Math. Soc., to appear.

[16]

D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654.  doi: 10.1090/S0894-0347-05-00484-4.

[17]

H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2), 162 (2005), 109-194.  doi: 10.4007/annals.2005.162.109.

[18]

N. Masmoudi and F. Rousset, Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equations, Arch. Ration. Mech. Anal., 223 (2017), 301-417.  doi: 10.1007/s00205-016-1036-5.

[19]

A. MorandoY. Trakhinin and P. Trebeschi, Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible MHD, Quart. Appl. Math., 72 (2014), 549-587.  doi: 10.1090/S0033-569X-2014-01346-7.

[20]

V. I. Nalimov, The Cauchy-Poisson problem, (Russian) Dinamika Splošn. Sredy Vyp. 18 Dinamika Židkost. so Svobod. Granicami., 254 (1974), 104–210.

[21]

P. Secchi and Y. Trakhinin, Well-posedness of the linearized plasma-vacuum interface problem, Interfaces Free Bound., 15 (2013), 323-357.  doi: 10.4171/IFB/305.

[22]

P. Secchi and Y. Trakhinin, Well-posedness of the plasma-vacuum interface problem, Nonlinearity, 27 (2014), 105-169.  doi: 10.1088/0951-7715/27/1/105.

[23]

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

[24]

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

[25]

Y. Sun, W. Wang and Z. Zhang, Well-posedness of the plasma-vacuum interface problem for ideal incompressible MHD, preprint, arXiv: 1705.00418.

[26]

M. Taylor, Partial Differential Equations, Vol. I-III, Berlin-Heidelberg-New York, Springer, 1996.1996.

[27]

Y. Trakhinin, The existence of current-vortex sheets in ideal compressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 191 (2009), 245-310.  doi: 10.1007/s00205-008-0124-6.

[28]

Y. Trakhinin, On the well-posedness of a linearized plasma-vacuum interface problem in ideal compressible MHD, J. Differential Equations, 249 (2010), 2577-2599.  doi: 10.1016/j.jde.2010.06.007.

[29]

Y. J. Wang and Z. Xin, Vanishing viscosity and surface tension limits of incompressible viscous surface waves, preprint, arXiv: 1504.00152. doi: 10.1007/s00220-014-1986-0.

[30]

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.

[31]

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.

[32]

S. Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135.  doi: 10.1007/s00222-009-0176-8.

[33]

S. Wu, Global wellposedness of the 3-D full water wave problem, Invent. Math., 184 (2011), 125-220.  doi: 10.1007/s00222-010-0288-1.

[34]

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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