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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

Yaobin Ou and  Pan Shi ,

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.

Keywords: Magnetohydrodynamics (MHD), free boundary problem, vacuum, global classical solutions, large initial data.
Mathematics Subject Classification: Primary: 35R35, 76N1.
Citation: Yaobin Ou, Pan Shi. Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 537-567. doi: 10.3934/dcdsb.2017026
References:
[1]

E. Becker, Gasdynamik, Teubner, Stuttgart, 1966.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

T.-P. Liu, Z. Xin and T. Yang, Vacuum states for compressible flow, Discrete and Continuous Dynamical Systems, 4 (1998), 1-32. 

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[41]

Y. -B. Zel'dovich and Y. -P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Vol. 2, Academic, New York, 1967.

[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.

[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.

[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.

show all references

References:
[1]

E. Becker, Gasdynamik, Teubner, Stuttgart, 1966.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

T.-P. Liu, Z. Xin and T. Yang, Vacuum states for compressible flow, Discrete and Continuous Dynamical Systems, 4 (1998), 1-32. 

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[41]

Y. -B. Zel'dovich and Y. -P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Vol. 2, Academic, New York, 1967.

[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.

[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.

[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.

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