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

Thermalization time in a model of neutron star

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.
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. 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. 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, (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. doi: 10.1006/aama.1995.1018. Google Scholar

[6]

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

[7]

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

[8]

E. Feireisl, Front propagation for degenerate parabolic equations,, Nonlinear Analysis, 35 (1999), 735. 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. doi: 10.1007/s002290050089. Google Scholar

[10]

M. Forestini, "Principes fondamentaux de structure stellaire,", Gordon and Breach, (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. 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. 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. doi: 10.1006/jdeq.1995.1111. Google Scholar

[14]

K. Jörgens, "Spectral Theory of Second-Order Ordinary Differential Operators,", Matematisk Institut, (1962). 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. 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. doi: 10.1007/BF02099452. Google Scholar

[17]

R. Kippenhahn and A. Weigert, "Stellar Structure and Evolution,", Springer Verlag, (1994). Google Scholar

[18]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", AMS Providence, (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. 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). doi: 10.1103/PhysRevC.75.065807. Google Scholar

[21]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Prentice-Hall Inc., (1967). Google Scholar

[22]

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

[23]

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

[24]

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

[25]

Z. Wu, J. Zhao, J. Yin and H. Li, "Nonlinear Diffusion Equations,", World Scientific, (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. 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. 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, (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. doi: 10.1006/aama.1995.1018. Google Scholar

[6]

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

[7]

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

[8]

E. Feireisl, Front propagation for degenerate parabolic equations,, Nonlinear Analysis, 35 (1999), 735. 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. doi: 10.1007/s002290050089. Google Scholar

[10]

M. Forestini, "Principes fondamentaux de structure stellaire,", Gordon and Breach, (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. 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. 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. doi: 10.1006/jdeq.1995.1111. Google Scholar

[14]

K. Jörgens, "Spectral Theory of Second-Order Ordinary Differential Operators,", Matematisk Institut, (1962). 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. 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. doi: 10.1007/BF02099452. Google Scholar

[17]

R. Kippenhahn and A. Weigert, "Stellar Structure and Evolution,", Springer Verlag, (1994). Google Scholar

[18]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", AMS Providence, (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. 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). doi: 10.1103/PhysRevC.75.065807. Google Scholar

[21]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Prentice-Hall Inc., (1967). Google Scholar

[22]

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

[23]

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

[24]

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

[25]

Z. Wu, J. Zhao, J. Yin and H. Li, "Nonlinear Diffusion Equations,", World Scientific, (2001). Google Scholar

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