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December 2017, 10(4): 1163-1203. doi: 10.3934/krm.2017045

Diffusive limit with geometric correction of unsteady neutron transport equation

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA

Received  October 2015 Revised  November 2016 Published  March 2017

We consider the diffusive limit of an unsteady neutron transportequation in a two-dimensional plate with one-speed velocity. We show the solution can be approximated by the sum of interior solution, initial layer, and boundary layer with geometric correction. Also, we construct a counterexample to the classical theory in [1] which states the behavior of solution near boundary can be described by the Knudsen layer derived from the Milne problem.

Citation: Lei Wu. Diffusive limit with geometric correction of unsteady neutron transport equation. Kinetic & Related Models, 2017, 10 (4) : 1163-1203. doi: 10.3934/krm.2017045
References:
[1]

A. BensoussanJ.-L. Lions and G. C. Papanicolaou, Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci., 15 (1979), 53-157. doi: 10.2977/prims/1195188427.

[2]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.

[3]

R. EspositoY. GuoC. Kim and R. Marra, Non-Isothermal boundary in the {Boltzmann} theory and {Fourier} law, Comm. Math. Phys., 323 (2013), 177-239. doi: 10.1007/s00220-013-1766-2.

[4]

E. W. Larsen, A functional-analytic approach to the steady, one-speed neutron transport equation with anisotropic scattering, Comm. Pure Appl. Math., 27 (1974), 523-545. doi: 10.1002/cpa.3160270404.

[5]

E. W. Larsen, Solutions of the steady, one-speed neutron transport equation for small mean free paths, J. Mathematical Phys., 15 (1974), 299-305. doi: 10.1063/1.1666642.

[6]

E. W. Larsen, Neutron transport and diffusion in inhomogeneous media Ⅰ, J. Mathematical Phys., 16 (1975), 1421-1427. doi: 10.1063/1.522714.

[7]

E. W. Larsen, Asymptotic theory of the linear transport equation for small mean free paths Ⅱ, SIAM J. Appl. Math., 33 (1977), 427-445. doi: 10.1137/0133027.

[8]

E. W. Larsen and J. D'Arruda, Asymptotic theory of the linear transport equation for small mean free paths Ⅰ, Phys. Rev., 13 (1976), 1933-1939. doi: 10.1103/PhysRevA.13.1933.

[9]

E. W. Larsen and G. J. Habetler, A functional-analytic derivation of Case's full and half-range formulas, Comm. Pure Appl. Math., 26 (1973), 525-537. doi: 10.1002/cpa.3160260406.

[10]

E. W. Larsen and J. B. Keller, Asymptotic solution of neutron transport problems for small mean free paths, J. Mathematical Phys., 15 (1974), 75-81. doi: 10.1063/1.1666510.

[11]

E. W. Larsen and P. F. Zweifel, On the spectrum of the linear transport operator, J. Mathematical Phys., 15 (1974), 1987-1997. doi: 10.1063/1.1666570.

[12]

E. W. Larsen and P. F. Zweifel, Steady, one-dimensional multigroup neutron transport with anisotropic scattering, J. Mathematical Phys., 17 (1976), 1812-1820. doi: 10.1063/1.522826.

[13]

L. Wu and Y. Guo, Geometric correction for diffusive expansion of steady neutron transport equation, Comm. Math. Phys., 226 (2015), 1473-1553. doi: 10.1007/s00220-015-2315-y.

show all references

References:
[1]

A. BensoussanJ.-L. Lions and G. C. Papanicolaou, Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci., 15 (1979), 53-157. doi: 10.2977/prims/1195188427.

[2]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.

[3]

R. EspositoY. GuoC. Kim and R. Marra, Non-Isothermal boundary in the {Boltzmann} theory and {Fourier} law, Comm. Math. Phys., 323 (2013), 177-239. doi: 10.1007/s00220-013-1766-2.

[4]

E. W. Larsen, A functional-analytic approach to the steady, one-speed neutron transport equation with anisotropic scattering, Comm. Pure Appl. Math., 27 (1974), 523-545. doi: 10.1002/cpa.3160270404.

[5]

E. W. Larsen, Solutions of the steady, one-speed neutron transport equation for small mean free paths, J. Mathematical Phys., 15 (1974), 299-305. doi: 10.1063/1.1666642.

[6]

E. W. Larsen, Neutron transport and diffusion in inhomogeneous media Ⅰ, J. Mathematical Phys., 16 (1975), 1421-1427. doi: 10.1063/1.522714.

[7]

E. W. Larsen, Asymptotic theory of the linear transport equation for small mean free paths Ⅱ, SIAM J. Appl. Math., 33 (1977), 427-445. doi: 10.1137/0133027.

[8]

E. W. Larsen and J. D'Arruda, Asymptotic theory of the linear transport equation for small mean free paths Ⅰ, Phys. Rev., 13 (1976), 1933-1939. doi: 10.1103/PhysRevA.13.1933.

[9]

E. W. Larsen and G. J. Habetler, A functional-analytic derivation of Case's full and half-range formulas, Comm. Pure Appl. Math., 26 (1973), 525-537. doi: 10.1002/cpa.3160260406.

[10]

E. W. Larsen and J. B. Keller, Asymptotic solution of neutron transport problems for small mean free paths, J. Mathematical Phys., 15 (1974), 75-81. doi: 10.1063/1.1666510.

[11]

E. W. Larsen and P. F. Zweifel, On the spectrum of the linear transport operator, J. Mathematical Phys., 15 (1974), 1987-1997. doi: 10.1063/1.1666570.

[12]

E. W. Larsen and P. F. Zweifel, Steady, one-dimensional multigroup neutron transport with anisotropic scattering, J. Mathematical Phys., 17 (1976), 1812-1820. doi: 10.1063/1.522826.

[13]

L. Wu and Y. Guo, Geometric correction for diffusive expansion of steady neutron transport equation, Comm. Math. Phys., 226 (2015), 1473-1553. doi: 10.1007/s00220-015-2315-y.

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