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October  2011, 16(3): 801-818. doi: 10.3934/dcdsb.2011.16.801

Thermalization time in a model of neutron star

Bernard Ducomet 1, and  Šárka Nečasová 2,

1. 

DPTA/Service de Physique Nucléaire, CEA, DAM, DIF, F-91297 Arpajon
2. 

Mathematical Institute, Academy of Sciences, Zitna 25, 11567 Prague 1

Received  April 2010 Revised  March 2011 Published  June 2011

We consider an initial boundary value problem for the equation describing heat conduction in a spherical model of neutron star considered by Lattimer et al. We estimate the asymptotic decay of the solution, which provides a plausible estimate for a "thermalization time" for the system.
Keywords: one-dimensional symmetry, Compressible, neutron star., heat conducting fluids.
Mathematics Subject Classification: 35Q30, 7676N1.
Citation: Bernard Ducomet, Šárka Nečasová. Thermalization time in a model of neutron star. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 801-818. doi: 10.3934/dcdsb.2011.16.801
References:
[1]

M. Bertsch, Asymptotic behavior of solutions of a nonlinear diffusion equation, SIAM J. Appl. Math., 42 (1982), 66-76. doi: 10.1137/0142005.  Google Scholar

[2]

J. G. Berryman and C. J. Holland, Asymptotic behavior of the nonlinear diffusion equation $n_t=(n^{-1}n_x)_x$ J. Math. Phys., 23 (1982), 983-987. doi: 10.1063/1.525466.  Google Scholar

[3]

S. Chandrasekhar, "An Introduction to the Study of Stellar Structures," Dover, 1967.  Google Scholar

[4]

H-Y. Chin, "Stellar Physics," Vol. 1, Blaisdell, Waltham, 1968. Google Scholar

[5]

S. Claudi and F. R. Guarguaglini, Large time behaviour for the heat equation with absorption and convection, Advances in Appl. Math., 16 (1995), 377-401. doi: 10.1006/aama.1995.1018.  Google Scholar

[6]

C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics," Springer-Verlag, Berlin Heidelberg, 2000.  Google Scholar

[7]

B. Ducomet and Š. Nečasová, On a fluid model of neutron star, Ann. Univ. Ferrara, 55 (2009), 153-193. doi: 10.1007/s11565-009-0067-3.  Google Scholar

[8]

E. Feireisl, Front propagation for degenerate parabolic equations, Nonlinear Analysis, 35 (1999), 735-746. doi: 10.1016/S0362-546X(98)00019-4.  Google Scholar

[9]

E. Feireisl and H. Petzeltová, On the zero-velocity limit solutions to the Navier-Stokes equations of compressible flow, Manuscripta Mathematica, 97 (1998), 109-116. doi: 10.1007/s002290050089.  Google Scholar

[10]

M. Forestini, "Principes fondamentaux de structure stellaire," Gordon and Breach, Overseas Publishers Association, 1999. Google Scholar

[11]

H. Fujita-Yashima and R. Benabidallah, Equation à symétrie sphérique d'un gaz visqueux et calorifère avec la surface libre, Annali di Matematica pura ed applicata, 168 (1995), 75-117. doi: 10.1007/BF01759255.  Google Scholar

[12]

M. A. Herrero and M. Pierre, The Cauchy problem for $u_t=\Delta u^m$ where $0 Transactions for the American Mathematical Society, 291 (1985), 145-158. doi: 10.2307/1999900.  Google Scholar

[13]

D. Hoff, Global solutions of the Navier - Stokes equations for multidimensional compressible flow with discontiuous initial data, J. Diff. Eqs., 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111.  Google Scholar

[14]

K. Jörgens, "Spectral Theory of Second-Order Ordinary Differential Operators," Matematisk Institut, Aarhus University, Lecture Notes Series N0. 2, 1962/63. Google Scholar

[15]

S. Jiang, On the asymptotic behavior of the motion of a viscous heat-conducting one-dimensional real, Math. Z., 190 (1994), 317-336. doi: 10.1007/BF02572324.  Google Scholar

[16]

S. Jiang, Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain, Commun. Math. Phys., 178 (1996), 339-374. doi: 10.1007/BF02099452.  Google Scholar

[17]

R. Kippenhahn and A. Weigert, "Stellar Structure and Evolution," Springer Verlag, Berlin-Heidelberg, 1994. Google Scholar

[18]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," AMS Providence, Rhode Island, 1968.  Google Scholar

[19]

J. M. Lattimer, K. A. Van Riper, M. Prakash and M. Prakash, Rapid cooling and the structure of neutron stars, The Astrophysical Journal, 425 (1994), 802-813. doi: 10.1086/174025.  Google Scholar

[20]

C. Monrozeau, J. Margueron and Ň. Sandulescu, Nuclear superfluidity and cooling time of neutron-star crusts, Physical Review, C75 (2007), 065807. doi: 10.1103/PhysRevC.75.065807.  Google Scholar

[21]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1967.  Google Scholar

[22]

G. Savaré and V. Vespri, The asymptotic profile of solutions of a class of doubly nonlinear equations, Nonlinear Analysis, Theory, Methods & Applications, 22 (1994) 1553-1565.  Google Scholar

[23]

J. L. Vazquez, "Smoothing and Decay Estimates for Nonlinear Diffusion Equations," Oxford University Press, Oxford, 2006.  Google Scholar

[24]

J. L. Vazquez, "The Porous Medium Equation," Clarendon Press, Oxford, 2007.  Google Scholar

[25]

Z. Wu, J. Zhao, J. Yin and H. Li, "Nonlinear Diffusion Equations," World Scientific, New Jersey, London, Singapore, Hong-Kong, 2001.  Google Scholar

show all references

References:
[1]

M. Bertsch, Asymptotic behavior of solutions of a nonlinear diffusion equation, SIAM J. Appl. Math., 42 (1982), 66-76. doi: 10.1137/0142005.  Google Scholar

[2]

J. G. Berryman and C. J. Holland, Asymptotic behavior of the nonlinear diffusion equation $n_t=(n^{-1}n_x)_x$ J. Math. Phys., 23 (1982), 983-987. doi: 10.1063/1.525466.  Google Scholar

[3]

S. Chandrasekhar, "An Introduction to the Study of Stellar Structures," Dover, 1967.  Google Scholar

[4]

H-Y. Chin, "Stellar Physics," Vol. 1, Blaisdell, Waltham, 1968. Google Scholar

[5]

S. Claudi and F. R. Guarguaglini, Large time behaviour for the heat equation with absorption and convection, Advances in Appl. Math., 16 (1995), 377-401. doi: 10.1006/aama.1995.1018.  Google Scholar

[6]

C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics," Springer-Verlag, Berlin Heidelberg, 2000.  Google Scholar

[7]

B. Ducomet and Š. Nečasová, On a fluid model of neutron star, Ann. Univ. Ferrara, 55 (2009), 153-193. doi: 10.1007/s11565-009-0067-3.  Google Scholar

[8]

E. Feireisl, Front propagation for degenerate parabolic equations, Nonlinear Analysis, 35 (1999), 735-746. doi: 10.1016/S0362-546X(98)00019-4.  Google Scholar

[9]

E. Feireisl and H. Petzeltová, On the zero-velocity limit solutions to the Navier-Stokes equations of compressible flow, Manuscripta Mathematica, 97 (1998), 109-116. doi: 10.1007/s002290050089.  Google Scholar

[10]

M. Forestini, "Principes fondamentaux de structure stellaire," Gordon and Breach, Overseas Publishers Association, 1999. Google Scholar

[11]

H. Fujita-Yashima and R. Benabidallah, Equation à symétrie sphérique d'un gaz visqueux et calorifère avec la surface libre, Annali di Matematica pura ed applicata, 168 (1995), 75-117. doi: 10.1007/BF01759255.  Google Scholar

[12]

M. A. Herrero and M. Pierre, The Cauchy problem for $u_t=\Delta u^m$ where $0 Transactions for the American Mathematical Society, 291 (1985), 145-158. doi: 10.2307/1999900.  Google Scholar

[13]

D. Hoff, Global solutions of the Navier - Stokes equations for multidimensional compressible flow with discontiuous initial data, J. Diff. Eqs., 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111.  Google Scholar

[14]

K. Jörgens, "Spectral Theory of Second-Order Ordinary Differential Operators," Matematisk Institut, Aarhus University, Lecture Notes Series N0. 2, 1962/63. Google Scholar

[15]

S. Jiang, On the asymptotic behavior of the motion of a viscous heat-conducting one-dimensional real, Math. Z., 190 (1994), 317-336. doi: 10.1007/BF02572324.  Google Scholar

[16]

S. Jiang, Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain, Commun. Math. Phys., 178 (1996), 339-374. doi: 10.1007/BF02099452.  Google Scholar

[17]

R. Kippenhahn and A. Weigert, "Stellar Structure and Evolution," Springer Verlag, Berlin-Heidelberg, 1994. Google Scholar

[18]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," AMS Providence, Rhode Island, 1968.  Google Scholar

[19]

J. M. Lattimer, K. A. Van Riper, M. Prakash and M. Prakash, Rapid cooling and the structure of neutron stars, The Astrophysical Journal, 425 (1994), 802-813. doi: 10.1086/174025.  Google Scholar

[20]

C. Monrozeau, J. Margueron and Ň. Sandulescu, Nuclear superfluidity and cooling time of neutron-star crusts, Physical Review, C75 (2007), 065807. doi: 10.1103/PhysRevC.75.065807.  Google Scholar

[21]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1967.  Google Scholar

[22]

G. Savaré and V. Vespri, The asymptotic profile of solutions of a class of doubly nonlinear equations, Nonlinear Analysis, Theory, Methods & Applications, 22 (1994) 1553-1565.  Google Scholar

[23]

J. L. Vazquez, "Smoothing and Decay Estimates for Nonlinear Diffusion Equations," Oxford University Press, Oxford, 2006.  Google Scholar

[24]

J. L. Vazquez, "The Porous Medium Equation," Clarendon Press, Oxford, 2007.  Google Scholar

[25]

Z. Wu, J. Zhao, J. Yin and H. Li, "Nonlinear Diffusion Equations," World Scientific, New Jersey, London, Singapore, Hong-Kong, 2001.  Google Scholar

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