# American Institute of Mathematical Sciences

January  2022, 27(1): 487-522. doi: 10.3934/dcdsb.2021052

## Global wellposedness of vacuum free boundary problem of isentropic compressible magnetohydrodynamic equations with axisymmetry

 School of Mathematics, Renmin University of China, Beijing 100872, China

* Corresponding author: ou@ruc.edu.cn

Received  November 2020 Published  January 2022 Early access  February 2021

In this paper, we prove the global existence of the strong solutions to the vacuum free boundary problem of isentropic compressible magnetohydrodynamic equations with small initial data and axial symmetry, where the solutions are independent of the axial variable and the angular variable. The solutions capture the precise physical behavior that the sound speed is $C^{1/2}$-Hölder continuous across the vacuum boundary provided that the adiabatic exponent $\gamma\in(1, 2)$. The main difficulties of this problem lie in the singularity at the symmetry axis, the degeneracy of the system near the free boundary and the strong coupling of the magnetic field and the velocity. We overcome the obstacles by constructing some new weighted nonlinear functionals (involving both lower-order and higher-order derivatives) and establishing the uniform-in-time weighted energy estimates of solutions by delicate analysis, in which the balance of pressure and self-gravitation, and the dissipation of velocity are crucial.

Citation: Kunquan Li, Yaobin Ou. Global wellposedness of vacuum free boundary problem of isentropic compressible magnetohydrodynamic equations with axisymmetry. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 487-522. doi: 10.3934/dcdsb.2021052
##### References:
 [1] G.-Q. Chen and D. Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376.  doi: 10.1006/jdeq.2001.4111. [2] D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary 1-D compressible Euler equations in physical vacuum, Comm. Pure Appl. Math., 64 (2011), 328-366.  doi: 10.1002/cpa.20344. [3] D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum, Arch. Ration. Mech. Anal., 206 (2012), 515-616.  doi: 10.1007/s00205-012-0536-1. [4] Y. Deng, T.-P. Liu, T. Yang and Z. Yao, Solutions of Euler-Poisson equations for gaseous stars, Arch. Ration. Mech. Anal., 164 (2002), 261-285.  doi: 10.1007/s00205-002-0209-6. [5] Q. Duan, Some Topics on Compressible Navier-Stokes Equations, Ph.D thesis, The Chinese University of Hong Kong (Hong Kong), 2011. [6] B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys., 266 (2006), 595-629.  doi: 10.1007/s00220-006-0052-y. [7] J. Fan, S. Huang and F. Li, Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum, Kinet. Relat. Models, 10 (2017), 1035-1053.  doi: 10.3934/krm.2017041. [8] J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409.  doi: 10.1016/j.nonrwa.2007.10.001. [9] P. Federbush, T. Luo and J. Smoller, Existence of magnetic compressible fluid stars, Arch. Ration. Mech. Anal., 215 (2015), 611-631.  doi: 10.1007/s00205-014-0790-5. [10] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004. [11] G. Gui, C. Wang and Y. Wang, Local well-posedness of the vacuum free boundary of 3-D compressible Navier-Stokes equations, Calc. Var. PDE, 58 (2019), Paper No. 166, 35 pp. doi: 10.1007/s00526-019-1608-y. [12] Z. Guo, H.-L. Li and Z. Xin, Lagrange structure and dynamics for solutions to the spherically symmetric compressible Navier-Stokes equations, Comm. Math. Phys., 309 (2012), 371-412.  doi: 10.1007/s00220-011-1334-6. [13] D. Hoff and H. K. Jenssen, Symmetric nonbarotropic flows with large data and forces, Arch. Ration. Mech. Anal., 173 (2004), 297-343.  doi: 10.1007/s00205-004-0318-5. [14] G. Hong, X. Hou, H. Peng and C. Zhu, Global existence for a class of large solutions to three-dimensional compressible magnetohydrodynamic equations with vacuum, SIAM J. Math. Anal., 49 (2017), 2409-2441.  doi: 10.1137/16M1100447. [15] G. Hong, T. Luo and C. Zhu, Global solutions to physical vacuum problem of non-isentropic viscous gaseous stars and nonlinear asymptotic stability of stationary solutions, J. Differential Equations, 265 (2018), 177-236.  doi: 10.1016/j.jde.2018.02.027. [16] Y. Hu and Q. Ju, Global large solutions of magnetohydrodynamics with temperature-dependent heat conductivity, Z. Angew. Math. Phys., 66 (2015), 865-889.  doi: 10.1007/s00033-014-0446-1. [17] X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284.  doi: 10.1007/s00220-008-0497-2. [18] J. Jang, Nonlinear instability in gravitational Euler-Poisson system for $\gamma = \frac 65$, Arch. Ration. Mech. Anal., 188 (2008), 265-307.  doi: 10.1007/s00205-007-0086-0. [19] J. Jang, Local well-posedness of dynamics of viscous gaseous stars, Arch. Ration. Mech. Anal., 195 (2010), 797-863.  doi: 10.1007/s00205-009-0253-6. [20] J. Jang, Nonlinear instability theory of Lane-Emden stars, Commun. Pure Appl. Math., 67 (2014), 1418-1465.  doi: 10.1002/cpa.21499. [21] J. Jang and N. Masmoudi, Well-posedness for compressible Euler with physical vacuum singularity, Commun. Pure Appl. Math., 62 (2009), 1327-1385.  doi: 10.1002/cpa.20285. [22] J. Jang and I. Tice, Instability theory of the Navier-Stokes-Poisson equations, Anal. PDE, 6 (2013), 1121-1181.  doi: 10.2140/apde.2013.6.1121. [23] J. Jang and N. Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, Commun. Pure Appl. Math., 68 (2015), 61-111.  doi: 10.1002/cpa.21517. [24] S. Jiang and J. Zhang, Boundary layers for the Navier-Stokes equations of compressible heat-conducting flows with cylindrical symmetry, SIAM J. Math. Anal., 41 (2009), 237-268.  doi: 10.1137/07070005X. [25] S. Jiang and J. Zhang, On the non-resistive limit and the magnetic boundary-layer for one-dimensional compressible magnetohydrodynamics, Nonlinearity, 30 (2017), 3587-3612.  doi: 10.1088/1361-6544/aa82f2. [26] A. Kufner, L. Maligranda and L.-E. Persson, The Hardy Inequality. About Its History and Some Related Results, Vy-davatelsksý Servis, Plzeň, 2007. [27] A.-G. Kulikovskiy and G.-A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, 1965. [28] L.-D. Laudau and E.-M. Lifshitz, Electrodynamics of Continuous Media, 2nd edn, Pergamon, New York, 1984. [29] Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.  doi: 10.1016/j.jde.2015.04.017. [30] K. Li, Z. Li and Y. Ou, Global axisymmetric classical solutions of full compressible magnetohydrodynamic equations with vacuum free boundary and large initial data, Sci. China Math., (2020). doi: 10.1007/s11425-019-1694-0. [31] H.-L. Li, X. Xu and J. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387.  doi: 10.1137/120893355. [32] S.-S. Lin, Stability of gaseous stars in spherically symmetric motions, SIAM J. Math. Anal., 28 (1997), 539-569.  doi: 10.1137/S0036141095292883. [33] X. Liu, Global solutions to compressible Navier-Stokes equations with spherical symmetry and free boundary, Nonlinear Anal. Real World Appl., 42 (2018), 220-254.  doi: 10.1016/j.nonrwa.2017.12.011. [34] X. Liu and Y. Yuan, Local existence and uniqueness of strong solutions to the free boundary problem of the full compressible Navier-Stokes equations in three dimensions, SIAM J. Math. Anal., 51 (2019), 748-789.  doi: 10.1137/18M1180426. [35] T. Luo, Some results on Newtonian gaseous stars-existence and stability, Acta Math. Appl. Sin. Engl. Ser., 35 (2019), 230-254.  doi: 10.1007/s10255-019-0804-z. [36] T. Luo, Z. Xin and H. Zeng, On nonlinear asymptotic stability of the Lane-Emden solutions for the viscous gaseous star problem, Adv. Math., 291 (2016), 90-182.  doi: 10.1016/j.aim.2015.12.022. [37] T. Luo, Z. Xin and H. Zeng, Nonlinear asymptotic stability of the Lane-Emden solutions for the viscous gaseous star problem with degenerate density dependent viscosities, Comm. Math. Phys., 347 (2016), 657-702.  doi: 10.1007/s00220-016-2753-1. [38] T. Luo, Z. Xin and H. Zeng, Well-posedness for the motion of physical vacuum of the three-dimensional compressible Euler equations with or without self-gravitation, Arch. Ration. Mech. Anal., 213 (2014), 763-831.  doi: 10.1007/s00205-014-0742-0. [39] Y. Ou, Low Mach and low Froude number limit for vacuum free boundary problem of all-time classical solutions of 1-D compressible Navier-Stokes equations, arXiv: 2004.04589. [40] Y. Ou and P. Shi, Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 537-567.  doi: 10.3934/dcdsb.2017026. [41] Y. Ou, P. Shi and P. Wittwer, Large time behaviors of strong solutions to magnetohydrodynamic equations with free boundary and degenerate viscosity, J. Math. Phys., 59 (2018), 081510, 34 pp. doi: 10.1063/1.5038584. [42] Y. Ou and H. Zeng, Global strong solutions to the vacuum free boundary problem for compressible Navier-Stokes equations with degenerate viscosity and gravity force, J. Differential Equations, 259 (2015), 6803-6829.  doi: 10.1016/j.jde.2015.08.008. [43] W. Su, Z. Guo and G. Yang, Global solution of 3D axially symmetric nonhomogeneous incompressible MHD equations, J. Differential Equations, 263 (2017), 8032-8073.  doi: 10.1016/j.jde.2017.08.035. [44] H. Wen and C. Zhu, Global symmetric classical solutions of the full compressible Navier-Stokes equations with vacuum and large initial data, J. Math. Pures Appl., 102 (2014), 498-545.  doi: 10.1016/j.matpur.2013.12.003. [45] H. Zeng, Global-in-time smoothness of solutions to the vacuum free boundary problem for compressible isentropic Navier-Stokes equations, Nonlinearity, 28 (2015), 331-345.  doi: 10.1088/0951-7715/28/2/331. [46] T. Zhang and D. Fang, Global behavior of spherically symmetric Navier-Stokes-Poisson system with degenerate viscosity coefficients, Arch. Ration. Mech. Anal., 191 (2009), 195-243.  doi: 10.1007/s00205-008-0183-8. [47] J. Zhang and F. Xie, Global solution for a one-dimensional model problem in thermally radiative magnetohydrodynamics, J. Differential Equations, 245 (2008), 1853-1882.  doi: 10.1016/j.jde.2008.07.010.

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##### References:
 [1] G.-Q. Chen and D. Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376.  doi: 10.1006/jdeq.2001.4111. [2] D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary 1-D compressible Euler equations in physical vacuum, Comm. Pure Appl. Math., 64 (2011), 328-366.  doi: 10.1002/cpa.20344. [3] D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum, Arch. Ration. Mech. Anal., 206 (2012), 515-616.  doi: 10.1007/s00205-012-0536-1. [4] Y. Deng, T.-P. Liu, T. Yang and Z. Yao, Solutions of Euler-Poisson equations for gaseous stars, Arch. Ration. Mech. Anal., 164 (2002), 261-285.  doi: 10.1007/s00205-002-0209-6. [5] Q. Duan, Some Topics on Compressible Navier-Stokes Equations, Ph.D thesis, The Chinese University of Hong Kong (Hong Kong), 2011. [6] B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys., 266 (2006), 595-629.  doi: 10.1007/s00220-006-0052-y. [7] J. Fan, S. Huang and F. Li, Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum, Kinet. Relat. Models, 10 (2017), 1035-1053.  doi: 10.3934/krm.2017041. [8] J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409.  doi: 10.1016/j.nonrwa.2007.10.001. [9] P. Federbush, T. Luo and J. Smoller, Existence of magnetic compressible fluid stars, Arch. Ration. Mech. Anal., 215 (2015), 611-631.  doi: 10.1007/s00205-014-0790-5. [10] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004. [11] G. Gui, C. Wang and Y. Wang, Local well-posedness of the vacuum free boundary of 3-D compressible Navier-Stokes equations, Calc. Var. PDE, 58 (2019), Paper No. 166, 35 pp. doi: 10.1007/s00526-019-1608-y. [12] Z. Guo, H.-L. Li and Z. Xin, Lagrange structure and dynamics for solutions to the spherically symmetric compressible Navier-Stokes equations, Comm. Math. Phys., 309 (2012), 371-412.  doi: 10.1007/s00220-011-1334-6. [13] D. Hoff and H. K. Jenssen, Symmetric nonbarotropic flows with large data and forces, Arch. Ration. Mech. Anal., 173 (2004), 297-343.  doi: 10.1007/s00205-004-0318-5. [14] G. Hong, X. Hou, H. Peng and C. Zhu, Global existence for a class of large solutions to three-dimensional compressible magnetohydrodynamic equations with vacuum, SIAM J. Math. Anal., 49 (2017), 2409-2441.  doi: 10.1137/16M1100447. [15] G. Hong, T. Luo and C. Zhu, Global solutions to physical vacuum problem of non-isentropic viscous gaseous stars and nonlinear asymptotic stability of stationary solutions, J. Differential Equations, 265 (2018), 177-236.  doi: 10.1016/j.jde.2018.02.027. [16] Y. Hu and Q. Ju, Global large solutions of magnetohydrodynamics with temperature-dependent heat conductivity, Z. Angew. Math. Phys., 66 (2015), 865-889.  doi: 10.1007/s00033-014-0446-1. [17] X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284.  doi: 10.1007/s00220-008-0497-2. [18] J. Jang, Nonlinear instability in gravitational Euler-Poisson system for $\gamma = \frac 65$, Arch. Ration. Mech. Anal., 188 (2008), 265-307.  doi: 10.1007/s00205-007-0086-0. [19] J. Jang, Local well-posedness of dynamics of viscous gaseous stars, Arch. Ration. Mech. Anal., 195 (2010), 797-863.  doi: 10.1007/s00205-009-0253-6. [20] J. Jang, Nonlinear instability theory of Lane-Emden stars, Commun. Pure Appl. Math., 67 (2014), 1418-1465.  doi: 10.1002/cpa.21499. [21] J. Jang and N. Masmoudi, Well-posedness for compressible Euler with physical vacuum singularity, Commun. Pure Appl. Math., 62 (2009), 1327-1385.  doi: 10.1002/cpa.20285. [22] J. Jang and I. Tice, Instability theory of the Navier-Stokes-Poisson equations, Anal. PDE, 6 (2013), 1121-1181.  doi: 10.2140/apde.2013.6.1121. [23] J. Jang and N. Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, Commun. Pure Appl. Math., 68 (2015), 61-111.  doi: 10.1002/cpa.21517. [24] S. Jiang and J. Zhang, Boundary layers for the Navier-Stokes equations of compressible heat-conducting flows with cylindrical symmetry, SIAM J. Math. Anal., 41 (2009), 237-268.  doi: 10.1137/07070005X. [25] S. Jiang and J. Zhang, On the non-resistive limit and the magnetic boundary-layer for one-dimensional compressible magnetohydrodynamics, Nonlinearity, 30 (2017), 3587-3612.  doi: 10.1088/1361-6544/aa82f2. [26] A. Kufner, L. Maligranda and L.-E. Persson, The Hardy Inequality. About Its History and Some Related Results, Vy-davatelsksý Servis, Plzeň, 2007. [27] A.-G. Kulikovskiy and G.-A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, 1965. [28] L.-D. Laudau and E.-M. Lifshitz, Electrodynamics of Continuous Media, 2nd edn, Pergamon, New York, 1984. [29] Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.  doi: 10.1016/j.jde.2015.04.017. [30] K. Li, Z. Li and Y. Ou, Global axisymmetric classical solutions of full compressible magnetohydrodynamic equations with vacuum free boundary and large initial data, Sci. China Math., (2020). doi: 10.1007/s11425-019-1694-0. [31] H.-L. Li, X. Xu and J. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387.  doi: 10.1137/120893355. [32] S.-S. Lin, Stability of gaseous stars in spherically symmetric motions, SIAM J. Math. Anal., 28 (1997), 539-569.  doi: 10.1137/S0036141095292883. [33] X. Liu, Global solutions to compressible Navier-Stokes equations with spherical symmetry and free boundary, Nonlinear Anal. Real World Appl., 42 (2018), 220-254.  doi: 10.1016/j.nonrwa.2017.12.011. [34] X. Liu and Y. Yuan, Local existence and uniqueness of strong solutions to the free boundary problem of the full compressible Navier-Stokes equations in three dimensions, SIAM J. Math. Anal., 51 (2019), 748-789.  doi: 10.1137/18M1180426. [35] T. Luo, Some results on Newtonian gaseous stars-existence and stability, Acta Math. Appl. Sin. Engl. Ser., 35 (2019), 230-254.  doi: 10.1007/s10255-019-0804-z. [36] T. Luo, Z. Xin and H. Zeng, On nonlinear asymptotic stability of the Lane-Emden solutions for the viscous gaseous star problem, Adv. Math., 291 (2016), 90-182.  doi: 10.1016/j.aim.2015.12.022. [37] T. Luo, Z. Xin and H. Zeng, Nonlinear asymptotic stability of the Lane-Emden solutions for the viscous gaseous star problem with degenerate density dependent viscosities, Comm. Math. Phys., 347 (2016), 657-702.  doi: 10.1007/s00220-016-2753-1. [38] T. Luo, Z. Xin and H. Zeng, Well-posedness for the motion of physical vacuum of the three-dimensional compressible Euler equations with or without self-gravitation, Arch. Ration. Mech. Anal., 213 (2014), 763-831.  doi: 10.1007/s00205-014-0742-0. [39] Y. Ou, Low Mach and low Froude number limit for vacuum free boundary problem of all-time classical solutions of 1-D compressible Navier-Stokes equations, arXiv: 2004.04589. [40] Y. Ou and P. Shi, Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 537-567.  doi: 10.3934/dcdsb.2017026. [41] Y. Ou, P. Shi and P. Wittwer, Large time behaviors of strong solutions to magnetohydrodynamic equations with free boundary and degenerate viscosity, J. Math. Phys., 59 (2018), 081510, 34 pp. doi: 10.1063/1.5038584. [42] Y. Ou and H. Zeng, Global strong solutions to the vacuum free boundary problem for compressible Navier-Stokes equations with degenerate viscosity and gravity force, J. Differential Equations, 259 (2015), 6803-6829.  doi: 10.1016/j.jde.2015.08.008. [43] W. Su, Z. Guo and G. Yang, Global solution of 3D axially symmetric nonhomogeneous incompressible MHD equations, J. Differential Equations, 263 (2017), 8032-8073.  doi: 10.1016/j.jde.2017.08.035. [44] H. Wen and C. Zhu, Global symmetric classical solutions of the full compressible Navier-Stokes equations with vacuum and large initial data, J. Math. Pures Appl., 102 (2014), 498-545.  doi: 10.1016/j.matpur.2013.12.003. [45] H. Zeng, Global-in-time smoothness of solutions to the vacuum free boundary problem for compressible isentropic Navier-Stokes equations, Nonlinearity, 28 (2015), 331-345.  doi: 10.1088/0951-7715/28/2/331. [46] T. Zhang and D. Fang, Global behavior of spherically symmetric Navier-Stokes-Poisson system with degenerate viscosity coefficients, Arch. Ration. Mech. Anal., 191 (2009), 195-243.  doi: 10.1007/s00205-008-0183-8. [47] J. Zhang and F. Xie, Global solution for a one-dimensional model problem in thermally radiative magnetohydrodynamics, J. Differential Equations, 245 (2008), 1853-1882.  doi: 10.1016/j.jde.2008.07.010.
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