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## 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  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021052
##### References:
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Math., 68 (2015), 61-111.  doi: 10.1002/cpa.21517.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] A. Kufner, L. Maligranda and L.-E. Persson, The Hardy Inequality. About Its History and Some Related Results, Vy-davatelsksý Servis, Plzeň, 2007.  Google Scholar [27] A.-G. Kulikovskiy and G.-A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, 1965. Google Scholar [28] L.-D. Laudau and E.-M. Lifshitz, Electrodynamics of Continuous Media, 2nd edn, Pergamon, New York, 1984. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [32] S.-S. Lin, Stability of gaseous stars in spherically symmetric motions, SIAM J. Math. Anal., 28 (1997), 539-569.  doi: 10.1137/S0036141095292883.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] Q. Duan, Some Topics on Compressible Navier-Stokes Equations, Ph.D thesis, The Chinese University of Hong Kong (Hong Kong), 2011.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] J. Jang, Nonlinear instability theory of Lane-Emden stars, Commun. Pure Appl. Math., 67 (2014), 1418-1465.  doi: 10.1002/cpa.21499.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] A. Kufner, L. Maligranda and L.-E. Persson, The Hardy Inequality. About Its History and Some Related Results, Vy-davatelsksý Servis, Plzeň, 2007.  Google Scholar [27] A.-G. Kulikovskiy and G.-A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, 1965. Google Scholar [28] L.-D. Laudau and E.-M. Lifshitz, Electrodynamics of Continuous Media, 2nd edn, Pergamon, New York, 1984. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [32] S.-S. Lin, Stability of gaseous stars in spherically symmetric motions, SIAM J. Math. Anal., 28 (1997), 539-569.  doi: 10.1137/S0036141095292883.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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. 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