# American Institute of Mathematical Sciences

March  2017, 22(2): 537-567. doi: 10.3934/dcdsb.2017026

## Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data

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

* Corresponding author

Received  March 2016 Revised  June 2016 Published  December 2016

The free boundary problem of planar full compressible magnetohydrodynamic equations with large initial data is studied in this paper, when the initial density connects to vacuum smoothly. The global existence and uniqueness of classical solutions are established, and the expanding rate of the free interface is shown. Using the method of Lagrangian particle path, we derive some L estimates and weighted energy estimates, which lead to the global existence of classical solutions. The main difficulty of this problem is the degeneracy of the system near the free boundary, while previous results (cf. [4,30]) require that the density is bounded from below by a positive constant.

Citation: Yaobin Ou, Pan Shi. Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 537-567. doi: 10.3934/dcdsb.2017026
##### References:
 [1] E. Becker, Gasdynamik, Teubner, Stuttgart, 1966.Google Scholar [2] D. Bian, B. Guo and J. Zhang, Global strong spherically symmetric solutions to the full compressible Navier-Stokes equations with stress free boundary J. Math. Phys. , 56 (2015), 023509, 23 pp.Google Scholar [3] G.-Q. Chen and M. Kratka, Global solutions to the Navier-Stokes equations for compressible heat conducting flow with symmetry and free boundary, Comm. Partial Diff. Eqs., 27 (2002), 907-943. doi: 10.1081/PDE-120004889. Google Scholar [4] G.-Q. Chen and D. Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Diff. Eqs., 182 (2002), 344-376. doi: 10.1006/jdeq.2001.4111. Google Scholar [5] D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary 1-D compressible Euler equations in physical vacuum, Commun. Pure Appl. Math., 64 (2011), 328-366. doi: 10.1002/cpa.20344. Google Scholar [6] 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 [7] Q. Duan, On the dynamics of Navier-Stokes equations for a shallow water model, J. Differential Equations, 250 (2011), 2687-2714. doi: 10.1016/j.jde.2011.01.010. Google Scholar [8] D. Fang and T. Zhang, Global behavior of compressible Navier-Stokes equations with a degenerate viscosity coefficient, Arch. Rational Mech. Anal., 182 (2006), 223-253. doi: 10.1007/s00205-006-0425-6. Google Scholar [9] D. Fang and T. Zhang, Global behavior of spherically symmetric Navier-Stokes-Poisson system with degenerate viscosity coefficients, Arch. Rational Mech. Anal., 191 (2009), 195-243. doi: 10.1007/s00205-008-0183-8. Google Scholar [10] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004. [11] Z. Guo, H. 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 [12] Z. Guo and Z. Li, Global existence of weak solution to the free boundary problem for compressible Navier-Stokes, Kinet. Relat. Models, 9 (2016), 75-103. doi: 10.3934/krm.2016.9.75. Google Scholar [13] Z. Guo and C. Zhu, Global weak solutions and asymptotic behavior to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum, J. Differential Equations, 248 (2010), 2768-2799. doi: 10.1016/j.jde.2010.03.005. Google Scholar [14] C. Hao, Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2885-2931. doi: 10.3934/dcdsb.2015.20.2885. Google Scholar [15] 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 [16] J. Jang, Local well-posedness of dynamics of viscous gaseous stars, Arch. Rational Mech. Anal., 195 (2010), 797-863. doi: 10.1007/s00205-009-0253-6. Google Scholar [17] J. Jang and N. Masmoudi, Well-posedness for compressible Euler equations with physical vacuum singularity, Commun. Pure Appl. Math., 62 (2009), 1327-1385. doi: 10.1002/cpa.20285. Google Scholar [18] 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 [19] S. Jiang, Z. Xin and P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity, Methods and Applications of Analysis, 12 (2005), 239-251. doi: 10.4310/MAA.2005.v12.n3.a2. Google Scholar [20] H. Li and X. Zhang, Global strong solutions to radial symmetric compressible Navier-Stokes equations with free boundary, J. Differential Equations, 261 (2016), 6341-6367. doi: 10.1016/j.jde.2016.08.038. Google Scholar [21] T.-P. Liu, Z. Xin and T. Yang, Vacuum states for compressible flow, Discrete and Continuous Dynamical Systems, 4 (1998), 1-32. Google Scholar [22] T.-P. Liu and T. Yang, Compressible flow with vacuum and physical singularity, Methods Appl. Anal., 7 (2000), 495-509. doi: 10.4310/MAA.2000.v7.n3.a7. Google Scholar [23] T. Luo, Z. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum, SIAM J. Math. Anal., 31 (2000), 1175-1191. doi: 10.1137/S0036141097331044. Google Scholar [24] T. Luo and H. Zeng, Global existence of smooth solutions and convergence to Barenblatt solutions for the physical vacuum free boundary problem of compressible Euler equations with damping, Communications on Pure and Applied Mathematics, 69 (2016), 1354-1396. doi: 10.1002/cpa.21562. Google Scholar [25] 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 [26] M. Okada, Free boundary problem for one-dimensional motions of compressible gas and vacuum, Japan J. Indust. Appl. Math., 21 (2004), 109-128. doi: 10.1007/BF03167467. Google Scholar [27] M. Okada and T. Makino, Free boundary problem for the equation of spherically symmetric motion of viscous gas, Japan J. Indust. Appl. Math., 10 (1993), 219-235. doi: 10.1007/BF03167573. Google Scholar [28] M. Okada, S. Matušč-Necasová and T. Makino, Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity, Ann. Univ. Ferrara Sez. Ⅶ (N.S.), 48 (2002), 1-20. Google Scholar [29] X. Qin and Z. Yao, Global smooth solutions of the compressible Navier-Stokes equations with density-dependent viscosity, J. Diff. Eqns., 244 (2008), 2041-2061. doi: 10.1016/j.jde.2007.11.001. Google Scholar [30] X. Qin and Z. Yao, Global solutions to planar magnetohydrodynamic equations with radiation and large initial data, Nonlinearity, 26 (2013), 591-619. doi: 10.1088/0951-7715/26/2/591. Google Scholar [31] Y. Qin, X. Liu and X. Yang, Global existence and exponential stability for a 1D compressible and radiative MHD flow, J. Differential Equations, 253 (2012), 1439-1488. doi: 10.1016/j.jde.2012.05.003. Google Scholar [32] J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. doi: 10.1137/0521061. Google Scholar [33] M. Umehara and A. Tani, Free-boundary problem of the one-dimensional equations for a viscous and heat-conductive gaseous flow under the self-gravitation, Math. Models Methods Appl. Sci., 23 (2013), 1377-1419. doi: 10.1142/S0218202513500127. Google Scholar [34] S. Vong, T. Yang and C. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Ⅱ, J. Diff. Eqs., 192 (2003), 475-501. doi: 10.1016/S0022-0396(03)00060-3. Google Scholar [35] D. Wang, On the global solution and interface behaviour of viscous compressible real flow with free boundaries, Nonlinearity, 16 (2003), 719-733. doi: 10.1088/0951-7715/16/2/321. Google Scholar [36] H. Wen and C. Zhu, Global Classical Large Solutions to Navier-Stokes Equations for Viscous Compressible and Heat-Conducting Fluids with Vacuum, SIAM J. Math. Anal., 45 (2013), 431-468. doi: 10.1137/120877829. Google Scholar [37] 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 [38] T. Yang, Z. Yao and C. Zhu, Compressible Navier-Stokes equations with density-dependent viscosity and vacuum, Comm. Partial Differential Equations, 26 (2001), 965-981. doi: 10.1081/PDE-100002385. Google Scholar [39] T. Yang and H. Zhao, A vacuum problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity, J. Diff. Eqs., 184 (2002), 163-184. doi: 10.1006/jdeq.2001.4140. Google Scholar [40] T. Yang and C. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum, Comm. Math. Phys., 230 (2002), 329-363. doi: 10.1007/s00220-002-0703-6. Google Scholar [41] Y. -B. Zel'dovich and Y. -P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Vol. 2, Academic, New York, 1967.Google Scholar [42] 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 [43] C. Zhu, Asymptotic behavior of compressible Navier-Stokes equations with density-dependent viscosity and vacuum, Comm. in Math. Phys., 293 (2010), 279-299. doi: 10.1007/s00220-009-0914-1. Google Scholar [44] C. Zhu and R. Zi, Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum, Discrete Contin. Dyn. Syst., 30 (2011), 1263-1283. doi: 10.3934/dcds.2011.30.1263. Google Scholar

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##### References:
 [1] E. Becker, Gasdynamik, Teubner, Stuttgart, 1966.Google Scholar [2] D. Bian, B. Guo and J. Zhang, Global strong spherically symmetric solutions to the full compressible Navier-Stokes equations with stress free boundary J. Math. Phys. , 56 (2015), 023509, 23 pp.Google Scholar [3] G.-Q. Chen and M. Kratka, Global solutions to the Navier-Stokes equations for compressible heat conducting flow with symmetry and free boundary, Comm. Partial Diff. Eqs., 27 (2002), 907-943. doi: 10.1081/PDE-120004889. Google Scholar [4] G.-Q. Chen and D. Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Diff. Eqs., 182 (2002), 344-376. doi: 10.1006/jdeq.2001.4111. Google Scholar [5] D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary 1-D compressible Euler equations in physical vacuum, Commun. Pure Appl. Math., 64 (2011), 328-366. doi: 10.1002/cpa.20344. Google Scholar [6] 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 [7] Q. Duan, On the dynamics of Navier-Stokes equations for a shallow water model, J. Differential Equations, 250 (2011), 2687-2714. doi: 10.1016/j.jde.2011.01.010. Google Scholar [8] D. Fang and T. Zhang, Global behavior of compressible Navier-Stokes equations with a degenerate viscosity coefficient, Arch. Rational Mech. Anal., 182 (2006), 223-253. doi: 10.1007/s00205-006-0425-6. Google Scholar [9] D. Fang and T. Zhang, Global behavior of spherically symmetric Navier-Stokes-Poisson system with degenerate viscosity coefficients, Arch. Rational Mech. Anal., 191 (2009), 195-243. doi: 10.1007/s00205-008-0183-8. Google Scholar [10] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004. [11] Z. Guo, H. 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 [12] Z. Guo and Z. Li, Global existence of weak solution to the free boundary problem for compressible Navier-Stokes, Kinet. Relat. Models, 9 (2016), 75-103. doi: 10.3934/krm.2016.9.75. Google Scholar [13] Z. Guo and C. Zhu, Global weak solutions and asymptotic behavior to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum, J. Differential Equations, 248 (2010), 2768-2799. doi: 10.1016/j.jde.2010.03.005. Google Scholar [14] C. Hao, Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2885-2931. doi: 10.3934/dcdsb.2015.20.2885. Google Scholar [15] 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 [16] J. Jang, Local well-posedness of dynamics of viscous gaseous stars, Arch. Rational Mech. Anal., 195 (2010), 797-863. doi: 10.1007/s00205-009-0253-6. Google Scholar [17] J. Jang and N. Masmoudi, Well-posedness for compressible Euler equations with physical vacuum singularity, Commun. Pure Appl. Math., 62 (2009), 1327-1385. doi: 10.1002/cpa.20285. Google Scholar [18] 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 [19] S. Jiang, Z. Xin and P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity, Methods and Applications of Analysis, 12 (2005), 239-251. doi: 10.4310/MAA.2005.v12.n3.a2. Google Scholar [20] H. Li and X. Zhang, Global strong solutions to radial symmetric compressible Navier-Stokes equations with free boundary, J. Differential Equations, 261 (2016), 6341-6367. doi: 10.1016/j.jde.2016.08.038. Google Scholar [21] T.-P. Liu, Z. Xin and T. Yang, Vacuum states for compressible flow, Discrete and Continuous Dynamical Systems, 4 (1998), 1-32. Google Scholar [22] T.-P. Liu and T. Yang, Compressible flow with vacuum and physical singularity, Methods Appl. Anal., 7 (2000), 495-509. doi: 10.4310/MAA.2000.v7.n3.a7. Google Scholar [23] T. Luo, Z. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum, SIAM J. Math. Anal., 31 (2000), 1175-1191. doi: 10.1137/S0036141097331044. Google Scholar [24] T. Luo and H. Zeng, Global existence of smooth solutions and convergence to Barenblatt solutions for the physical vacuum free boundary problem of compressible Euler equations with damping, Communications on Pure and Applied Mathematics, 69 (2016), 1354-1396. doi: 10.1002/cpa.21562. Google Scholar [25] 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 [26] M. Okada, Free boundary problem for one-dimensional motions of compressible gas and vacuum, Japan J. Indust. Appl. Math., 21 (2004), 109-128. doi: 10.1007/BF03167467. Google Scholar [27] M. Okada and T. Makino, Free boundary problem for the equation of spherically symmetric motion of viscous gas, Japan J. Indust. Appl. Math., 10 (1993), 219-235. doi: 10.1007/BF03167573. Google Scholar [28] M. Okada, S. Matušč-Necasová and T. Makino, Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity, Ann. Univ. Ferrara Sez. Ⅶ (N.S.), 48 (2002), 1-20. Google Scholar [29] X. Qin and Z. Yao, Global smooth solutions of the compressible Navier-Stokes equations with density-dependent viscosity, J. Diff. Eqns., 244 (2008), 2041-2061. doi: 10.1016/j.jde.2007.11.001. Google Scholar [30] X. Qin and Z. Yao, Global solutions to planar magnetohydrodynamic equations with radiation and large initial data, Nonlinearity, 26 (2013), 591-619. doi: 10.1088/0951-7715/26/2/591. Google Scholar [31] Y. Qin, X. Liu and X. Yang, Global existence and exponential stability for a 1D compressible and radiative MHD flow, J. Differential Equations, 253 (2012), 1439-1488. doi: 10.1016/j.jde.2012.05.003. Google Scholar [32] J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. doi: 10.1137/0521061. Google Scholar [33] M. Umehara and A. Tani, Free-boundary problem of the one-dimensional equations for a viscous and heat-conductive gaseous flow under the self-gravitation, Math. Models Methods Appl. Sci., 23 (2013), 1377-1419. doi: 10.1142/S0218202513500127. Google Scholar [34] S. Vong, T. Yang and C. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Ⅱ, J. Diff. Eqs., 192 (2003), 475-501. doi: 10.1016/S0022-0396(03)00060-3. Google Scholar [35] D. Wang, On the global solution and interface behaviour of viscous compressible real flow with free boundaries, Nonlinearity, 16 (2003), 719-733. doi: 10.1088/0951-7715/16/2/321. Google Scholar [36] H. Wen and C. Zhu, Global Classical Large Solutions to Navier-Stokes Equations for Viscous Compressible and Heat-Conducting Fluids with Vacuum, SIAM J. Math. Anal., 45 (2013), 431-468. doi: 10.1137/120877829. Google Scholar [37] 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 [38] T. Yang, Z. Yao and C. Zhu, Compressible Navier-Stokes equations with density-dependent viscosity and vacuum, Comm. Partial Differential Equations, 26 (2001), 965-981. doi: 10.1081/PDE-100002385. Google Scholar [39] T. Yang and H. Zhao, A vacuum problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity, J. Diff. Eqs., 184 (2002), 163-184. doi: 10.1006/jdeq.2001.4140. Google Scholar [40] T. Yang and C. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum, Comm. Math. Phys., 230 (2002), 329-363. doi: 10.1007/s00220-002-0703-6. Google Scholar [41] Y. -B. Zel'dovich and Y. -P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Vol. 2, Academic, New York, 1967.Google Scholar [42] 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 [43] C. Zhu, Asymptotic behavior of compressible Navier-Stokes equations with density-dependent viscosity and vacuum, Comm. in Math. Phys., 293 (2010), 279-299. doi: 10.1007/s00220-009-0914-1. Google Scholar [44] C. Zhu and R. Zi, Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum, Discrete Contin. Dyn. Syst., 30 (2011), 1263-1283. doi: 10.3934/dcds.2011.30.1263. Google Scholar
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