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 & Applied Analysis, 2019, 18 (2) : 569-602. doi: 10.3934/cpaa.2019029
References:
[1]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

[2]

Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021028

[3]

Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387

[4]

Huijuan Song, Bei Hu, Zejia Wang. Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 667-691. doi: 10.3934/dcdsb.2020084

[5]

Lei Yang, Lianzhang Bao. Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1083-1109. doi: 10.3934/dcdsb.2020154

[6]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004

[7]

Yao Nie, Jia Yuan. The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020397

[8]

Min Xi, Wenyu Sun, Jun Chen. Survey of derivative-free optimization. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 537-555. doi: 10.3934/naco.2020050

[9]

Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377

[10]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262

[11]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115

[12]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020390

[13]

Claude-Michel Brauner, Luca Lorenzi. Instability of free interfaces in premixed flame propagation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 575-596. doi: 10.3934/dcdss.2020363

[14]

Aurelia Dymek. Proximality of multidimensional $ \mathscr{B} $-free systems. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021013

[15]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163

[16]

Rong Wang, Yihong Du. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020360

[17]

Maoli Chen, Xiao Wang, Yicheng Liu. Collision-free flocking for a time-delay system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1223-1241. doi: 10.3934/dcdsb.2020251

[18]

Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039

[19]

Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002

[20]

Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (101)
  • HTML views (179)
  • Cited by (0)

Other articles
by authors

[Back to Top]