September  2019, 18(5): 2529-2574. doi: 10.3934/cpaa.2019115

Global existence and asymptotic behavior of spherically symmetric solutions for the multi-dimensional infrarelativistic model

1. 

Department of Applied Mathematics, Donghua University, Shanghai 201620, China

2. 

Department of Applied Mathematics, College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China

* Corresponding author

Received  July 2018 Revised  July 2018 Published  April 2019

Fund Project: The first author is supported by NSF grant 11671075, and the second author is supported by NSF grant 11801133 and the grant from the Key Research Projects of He'nan Higher Education Institutions (18A110038).

In this paper, we establish the global existence, uniqueness and asymptotic behavior of spherically symmetric solutions for the multi-dimensional infrarelativistic model in $ H^i\times H^i\times H^i\times H^{i+1}\;(i = 1,2,4) $.

Citation: Yuming Qin, Jianlin Zhang. Global existence and asymptotic behavior of spherically symmetric solutions for the multi-dimensional infrarelativistic model. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2529-2574. doi: 10.3934/cpaa.2019115
References:
[1]

F. AzevedoE. Sauter and M. Thompson, Classical global solutions for compressible radiation flow in a slab under semi-reflexive boundary conditions, Nonlinear Analysis: Real World Applications, 37 (2017), 493-511.  doi: 10.1016/j.nonrwa.2017.03.007.

[2]

C. Buet and B. Despres, Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, J. Quant. Spectroscopy Radiative Transfer, 85 (2004), 385-418. 

[3]

S. Chandrasekhar, Radiative Transfer, Dover Publications, Inc, New York, 1960.

[4]

B. Dubroca and J. L. Feugeas, Etude théorique et numérique d'une hiérarchie de modèles aux moments pour le transfert radiatif, Comptes Rendus de I'Academie des Sciences, Série I, 329 (1999), 915-920.  doi: 10.1016/S0764-4442(00)87499-6.

[5]

B. Ducomet and S. Nečasová, Global existence of solutions for the one-dimensional motions of a compressible viscous gas with radiation: An "infrarelativistic model", Nonlinear Analysis TMA, 72 (2010), 3258-3274.  doi: 10.1016/j.na.2009.12.005.

[6]

B. Ducomet and S. Nečasová, Large-time behavior of the motion of a viscous heat-conducting one-dimensional gas coupled to radiation, Annali di Matematica, 191 (2012), 219-260.  doi: 10.1007/s10231-010-0180-z.

[7]

B. Ducomet and S. Nečasová, Asymptotic behavior of the motion of a viscous heat-conducting one-dimensional gas with radiation: the pure scattering case, Analysis and Applications, 11 (2013), 1350003(29 pages). doi: 10.1142/S0219530513500036.

[8]

F. Golse and B. Perthame, Generalized solutions of the radiative transfer equations in a singular case, Comm. Math. Physics, 106 (1986), 211-239. 

[9]

Z. GuoH. Li and Z. Xin, Lagrange structure and dynamics for solutions to the spherically symmetric compressible Navier-Stokes equations, Commun. Math. Phys., 309 (2012), 371-412.  doi: 10.1007/s00220-011-1334-6.

[10]

D. Hoff, Spherically symmetric solutions of the Navier-Stokes equations for compressible, isothermal flow with large, discontinuous initial data, Indiana Univ. Math. J., 41 (1992), 1225-1302.  doi: 10.1512/iumj.1992.41.41060.

[11]

S. Jiang, Global spherically symmetric solutions of the equations of a viscous polytropic ideal gas in an exterior domain, Commun. Math. Phys., 178 (1996), 339-374. 

[12]

S. Jiang and P. Zhang, Global spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Commun. Math. Phys., 215 (2001), 559-581.  doi: 10.1007/PL00005543.

[13]

Y. Li and S. Zhu, Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum, J. Differential Equations, 256 (2014), 3943-3980.  doi: 10.1016/j.jde.2014.03.007.

[14]

C. Lin, Mathematical Analysis of Radiative Transfer Models, Ph.D Thesis, Université des Sciences et Technologies de Lille, 2007.

[15]

C. LinJ. F. Coulombel and T. Goudon, Shock profiles for non-equilibrium radiating gases, Physics D, 218 (2006), 83-94.  doi: 10.1016/j.physd.2006.04.012.

[16]

R. B. LowrieJ. E. Morel and J. A. Hittinger, The coupling of radiation and hydrodynamics, Astrophysical Journal, 521 (1999), 432-450. 

[17] D. Mihalas and B. W. Mihalas, Foundations of Radiation Hydrodynamics, Oxford Univ. Press, New York, 1984. 
[18] G. C. Pomraning, The Equations of Radiation Hydrodynamics, Pergamon Press, New York, 1973. 
[19]

G. C. Pomraning, Radiation Hydrodynamics, Dover Publications, Inc, New York, 2005.

[20]

Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, Volume 184, Advances in Partial Differential Equations, Birkhauser Verlag AG, Basel-Boston-Berlin, 2008.

[21]

Y. QinB. Feng and M. Zhang, Large-time behavior of solutions for the one-dimensional infrarelativistic model of a compressible viscous gas with radiation, J. Differential Equations, 252 (2012), 6175-6213.  doi: 10.1016/j.jde.2012.02.022.

[22]

Y. Qin, B. Feng and M. Zhang, Large-time behavior of solutions for the 1D viscous heat-conducting gas with radiation: the pure scattering case, J. Differential Equations, 256 (2014), 989–1042. doi: 10.1016/j.jde.2013.10.003.

[23]

Y. Qin and L. Huang, Global well-Posedness of Nonlinear Parabolic-Hyperbolic Coupled Systems, Frontiers in Mathematics, Springer Basel AG, 2012. doi: 10.1007/978-3-0348-0280-2.

[24]

Y. Qin and J. Zhang, Global existence and asymptotic behavior of cylindrically symmetric solutions for the 3D infrarelativistic model with radiation, J. Math. Fluid Mech., 1 (2018), 35-81.  doi: 10.1007/s00021-016-0312-3.

[25]

S. F. Shandarin and Y. B. Zel'dovichi, The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium, Rev. Modern Phys., 61 (1989), 185-220.  doi: 10.1103/RevModPhys.61.185.

[26]

M. Umehara and A. Tani, Global solution to the one-dimensional equations for a selfgravitating viscous radiative and reactive gas, J. Differential Equations, 234 (2007), 439-463. doi: 10.1016/j.jde.2006.09.023.

[27]

M. Umehara and A. Tani, Temporally global solution to the equations for a spherically symmetric viscous radiative and reactive gas over the rigid core, Anal. Appl., 6 (2008), 183-211.  doi: 10.1142/S0219530508001122.

[28]

Z. Xin and H. Yuan, Vacuum state for spherically symmetric solutions of the compressible Navier-Stokes equations, J. Hyperbolic Differential Equations, 3 (2006), 403-442.  doi: 10.1142/S0219891606000847.

[29]

X. Zhang and S. Jiang, Local existence and finite-time blow-up in multidimensional radiation hydrodynamics, J. Math. Fluid Mech., 9 (2007), 543-564.  doi: 10.1007/s00021-005-0213-3.

show all references

References:
[1]

F. AzevedoE. Sauter and M. Thompson, Classical global solutions for compressible radiation flow in a slab under semi-reflexive boundary conditions, Nonlinear Analysis: Real World Applications, 37 (2017), 493-511.  doi: 10.1016/j.nonrwa.2017.03.007.

[2]

C. Buet and B. Despres, Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, J. Quant. Spectroscopy Radiative Transfer, 85 (2004), 385-418. 

[3]

S. Chandrasekhar, Radiative Transfer, Dover Publications, Inc, New York, 1960.

[4]

B. Dubroca and J. L. Feugeas, Etude théorique et numérique d'une hiérarchie de modèles aux moments pour le transfert radiatif, Comptes Rendus de I'Academie des Sciences, Série I, 329 (1999), 915-920.  doi: 10.1016/S0764-4442(00)87499-6.

[5]

B. Ducomet and S. Nečasová, Global existence of solutions for the one-dimensional motions of a compressible viscous gas with radiation: An "infrarelativistic model", Nonlinear Analysis TMA, 72 (2010), 3258-3274.  doi: 10.1016/j.na.2009.12.005.

[6]

B. Ducomet and S. Nečasová, Large-time behavior of the motion of a viscous heat-conducting one-dimensional gas coupled to radiation, Annali di Matematica, 191 (2012), 219-260.  doi: 10.1007/s10231-010-0180-z.

[7]

B. Ducomet and S. Nečasová, Asymptotic behavior of the motion of a viscous heat-conducting one-dimensional gas with radiation: the pure scattering case, Analysis and Applications, 11 (2013), 1350003(29 pages). doi: 10.1142/S0219530513500036.

[8]

F. Golse and B. Perthame, Generalized solutions of the radiative transfer equations in a singular case, Comm. Math. Physics, 106 (1986), 211-239. 

[9]

Z. GuoH. Li and Z. Xin, Lagrange structure and dynamics for solutions to the spherically symmetric compressible Navier-Stokes equations, Commun. Math. Phys., 309 (2012), 371-412.  doi: 10.1007/s00220-011-1334-6.

[10]

D. Hoff, Spherically symmetric solutions of the Navier-Stokes equations for compressible, isothermal flow with large, discontinuous initial data, Indiana Univ. Math. J., 41 (1992), 1225-1302.  doi: 10.1512/iumj.1992.41.41060.

[11]

S. Jiang, Global spherically symmetric solutions of the equations of a viscous polytropic ideal gas in an exterior domain, Commun. Math. Phys., 178 (1996), 339-374. 

[12]

S. Jiang and P. Zhang, Global spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Commun. Math. Phys., 215 (2001), 559-581.  doi: 10.1007/PL00005543.

[13]

Y. Li and S. Zhu, Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum, J. Differential Equations, 256 (2014), 3943-3980.  doi: 10.1016/j.jde.2014.03.007.

[14]

C. Lin, Mathematical Analysis of Radiative Transfer Models, Ph.D Thesis, Université des Sciences et Technologies de Lille, 2007.

[15]

C. LinJ. F. Coulombel and T. Goudon, Shock profiles for non-equilibrium radiating gases, Physics D, 218 (2006), 83-94.  doi: 10.1016/j.physd.2006.04.012.

[16]

R. B. LowrieJ. E. Morel and J. A. Hittinger, The coupling of radiation and hydrodynamics, Astrophysical Journal, 521 (1999), 432-450. 

[17] D. Mihalas and B. W. Mihalas, Foundations of Radiation Hydrodynamics, Oxford Univ. Press, New York, 1984. 
[18] G. C. Pomraning, The Equations of Radiation Hydrodynamics, Pergamon Press, New York, 1973. 
[19]

G. C. Pomraning, Radiation Hydrodynamics, Dover Publications, Inc, New York, 2005.

[20]

Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, Volume 184, Advances in Partial Differential Equations, Birkhauser Verlag AG, Basel-Boston-Berlin, 2008.

[21]

Y. QinB. Feng and M. Zhang, Large-time behavior of solutions for the one-dimensional infrarelativistic model of a compressible viscous gas with radiation, J. Differential Equations, 252 (2012), 6175-6213.  doi: 10.1016/j.jde.2012.02.022.

[22]

Y. Qin, B. Feng and M. Zhang, Large-time behavior of solutions for the 1D viscous heat-conducting gas with radiation: the pure scattering case, J. Differential Equations, 256 (2014), 989–1042. doi: 10.1016/j.jde.2013.10.003.

[23]

Y. Qin and L. Huang, Global well-Posedness of Nonlinear Parabolic-Hyperbolic Coupled Systems, Frontiers in Mathematics, Springer Basel AG, 2012. doi: 10.1007/978-3-0348-0280-2.

[24]

Y. Qin and J. Zhang, Global existence and asymptotic behavior of cylindrically symmetric solutions for the 3D infrarelativistic model with radiation, J. Math. Fluid Mech., 1 (2018), 35-81.  doi: 10.1007/s00021-016-0312-3.

[25]

S. F. Shandarin and Y. B. Zel'dovichi, The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium, Rev. Modern Phys., 61 (1989), 185-220.  doi: 10.1103/RevModPhys.61.185.

[26]

M. Umehara and A. Tani, Global solution to the one-dimensional equations for a selfgravitating viscous radiative and reactive gas, J. Differential Equations, 234 (2007), 439-463. doi: 10.1016/j.jde.2006.09.023.

[27]

M. Umehara and A. Tani, Temporally global solution to the equations for a spherically symmetric viscous radiative and reactive gas over the rigid core, Anal. Appl., 6 (2008), 183-211.  doi: 10.1142/S0219530508001122.

[28]

Z. Xin and H. Yuan, Vacuum state for spherically symmetric solutions of the compressible Navier-Stokes equations, J. Hyperbolic Differential Equations, 3 (2006), 403-442.  doi: 10.1142/S0219891606000847.

[29]

X. Zhang and S. Jiang, Local existence and finite-time blow-up in multidimensional radiation hydrodynamics, J. Math. Fluid Mech., 9 (2007), 543-564.  doi: 10.1007/s00021-005-0213-3.

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