# American Institute of Mathematical Sciences

December  2013, 6(4): 671-686. doi: 10.3934/krm.2013.6.671

## A Milne problem from a Bose condensate with excitations

 1 Mathematical Sciences, 41296 Göteborg, Sweden 2 LATP, Aix-Marseille University, France

Received  March 2013 Revised  June 2013 Published  November 2013

This paper deals with a half-space linearized problem for the distribution function of the excitations in a Bose gas close to equilibrium. Existence and uniqueness of the solution, as well as its asymptotic properties are proven for a given energy flow. The problem differs from the ones for the classical Boltzmann and related equations, where the hydrodynamic mass flow along the half-line is constant. Here it is no more constant. Instead we use the energy flow which is constant, but no more hydrodynamic.
Citation: Leif Arkeryd, Anne Nouri. A Milne problem from a Bose condensate with excitations. Kinetic & Related Models, 2013, 6 (4) : 671-686. doi: 10.3934/krm.2013.6.671
##### References:
 [1] L. Arkeryd and A. Nouri, Bose condensates in interaction with excitations - a kinetic model, Commun. Math. Phys., 310 (2012), 765-788. doi: 10.1007/s00220-012-1415-1.  Google Scholar [2] L. Arkeryd and A. Nouri, Bose Condensates in Interaction with Excitations - A Two-Component, Space Dependent Model Close to Equilibrium,, in preparation., ().   Google Scholar [3] L. Arkeryd and A. Nouri, On the Milne problem and the hydrodynamic limit for a steady Boltzmann equation model, J. Stat. Phys., 99 (2000), 993-1019. doi: 10.1023/A:1018655815285.  Google Scholar [4] A. V. Bobylev and N. Bernhoff, Discrete velocity models and dynamic systems, in Lecture Notes on the discretization of the Boltzmann equation (eds. World Sci. Pub. I), 63 (2003), 203-222. doi: 10.1142/9789812796905_0008.  Google Scholar [5] C. Bardos, R. E. Caflish and B. Nicolaenko, The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas, Commun. Pure Appl. Math., 39 (1986), 323-352. doi: 10.1002/cpa.3160390304.  Google Scholar [6] C. Bardos, F. Golse and Y. Sone, Half-space problems for the Boltzmann equation: A survey, J. Stat. Phys., 124 (2006), 275-300. doi: 10.1007/s10955-006-9077-z.  Google Scholar [7] A. V. Bobylev and G. Toscani, Two-dimensional half space problems for the Broadwell discrete velocity model, Contin. Mech. Thermodyn., 8 (1996), 257-274. doi: 10.1007/s001610050043.  Google Scholar [8] C. Cercignani, Half-space problems in the kinetic theory of gases, Trends in applications of pure mathematics to mechanics (Bad Honnef, 1985), in Lect. Notes Phys., Springer-Verlag New-York, 249 (1986), 35-50. doi: 10.1007/BFb0016381.  Google Scholar [9] F. Coron, F. Golse and C. Sulem, A classification of well-posed kinetic layer problems, Commun. Pure Appl. Math., 41 (1988), 409-435. doi: 10.1002/cpa.3160410403.  Google Scholar [10] C. Cercignani, R. Marra and R. Esposito, The Milne problem with a force term, Transport Theory and Statistical Physics, 27 (1998), 1-33. doi: 10.1080/00411459808205139.  Google Scholar [11] F. Golse and F. Poupaud, Stationary solutions of the linearized Boltzmann equation in a half-space, Math. Methods Appl. Sci., 11 (1989), 483-502 . doi: 10.1002/mma.1670110406.  Google Scholar [12] N. Maslova, The Kramers problems in the kinetic theory of gases, USSR Comput. Math. Phys., 22 (1982), 208-219. Google Scholar [13] N. Maslova, Nonlinear Evolution Equations, Kinetic approach. Series on Advances in Mathematics for Applied Sciences, 10. World Scientific Publishing Co., Inc., River Edge, NJ, 1993.  Google Scholar [14] F. Poupaud, Diffusion approximation of the linear semiconductor equation: analysis of boundary layers, Asymptotic Analysis, 4 (1991), 293-317.  Google Scholar [15] Y. Sone, Kinetic Theory and Fluid Dynamics, Birkhauser Boston, 2002. Google Scholar [16] Y. Sone, Molecular Gas Dynamics, Theory, techniques, and applications. Modeling and Simulation in Science, Engineering and Technology. Birkh?user Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4573-1.  Google Scholar [17] S. Ukai, T. Yang and S.-H. Yu, Nonlinear boundary layers of the Boltzmann equation: I. Existence, Commun. Math. Phys., 236 (2003), 373-393. doi: 10.1007/s00220-003-0822-8.  Google Scholar [18] S. Ukai, T. Yang and S.-H. Yu, Nonlinear stability of boundary layers of the Boltzmann equation; I. The case $\mathcalM_\infty <-1$, Commun. Math. Phys., 244 (2004), 99-109. doi: 10.1007/s00220-003-0976-4.  Google Scholar

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##### References:
 [1] L. Arkeryd and A. Nouri, Bose condensates in interaction with excitations - a kinetic model, Commun. Math. Phys., 310 (2012), 765-788. doi: 10.1007/s00220-012-1415-1.  Google Scholar [2] L. Arkeryd and A. Nouri, Bose Condensates in Interaction with Excitations - A Two-Component, Space Dependent Model Close to Equilibrium,, in preparation., ().   Google Scholar [3] L. Arkeryd and A. Nouri, On the Milne problem and the hydrodynamic limit for a steady Boltzmann equation model, J. Stat. Phys., 99 (2000), 993-1019. doi: 10.1023/A:1018655815285.  Google Scholar [4] A. V. Bobylev and N. Bernhoff, Discrete velocity models and dynamic systems, in Lecture Notes on the discretization of the Boltzmann equation (eds. World Sci. Pub. I), 63 (2003), 203-222. doi: 10.1142/9789812796905_0008.  Google Scholar [5] C. Bardos, R. E. Caflish and B. Nicolaenko, The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas, Commun. Pure Appl. Math., 39 (1986), 323-352. doi: 10.1002/cpa.3160390304.  Google Scholar [6] C. Bardos, F. Golse and Y. Sone, Half-space problems for the Boltzmann equation: A survey, J. Stat. Phys., 124 (2006), 275-300. doi: 10.1007/s10955-006-9077-z.  Google Scholar [7] A. V. Bobylev and G. Toscani, Two-dimensional half space problems for the Broadwell discrete velocity model, Contin. Mech. Thermodyn., 8 (1996), 257-274. doi: 10.1007/s001610050043.  Google Scholar [8] C. Cercignani, Half-space problems in the kinetic theory of gases, Trends in applications of pure mathematics to mechanics (Bad Honnef, 1985), in Lect. Notes Phys., Springer-Verlag New-York, 249 (1986), 35-50. doi: 10.1007/BFb0016381.  Google Scholar [9] F. Coron, F. Golse and C. Sulem, A classification of well-posed kinetic layer problems, Commun. Pure Appl. Math., 41 (1988), 409-435. doi: 10.1002/cpa.3160410403.  Google Scholar [10] C. Cercignani, R. Marra and R. Esposito, The Milne problem with a force term, Transport Theory and Statistical Physics, 27 (1998), 1-33. doi: 10.1080/00411459808205139.  Google Scholar [11] F. Golse and F. Poupaud, Stationary solutions of the linearized Boltzmann equation in a half-space, Math. Methods Appl. Sci., 11 (1989), 483-502 . doi: 10.1002/mma.1670110406.  Google Scholar [12] N. Maslova, The Kramers problems in the kinetic theory of gases, USSR Comput. Math. Phys., 22 (1982), 208-219. Google Scholar [13] N. Maslova, Nonlinear Evolution Equations, Kinetic approach. Series on Advances in Mathematics for Applied Sciences, 10. World Scientific Publishing Co., Inc., River Edge, NJ, 1993.  Google Scholar [14] F. Poupaud, Diffusion approximation of the linear semiconductor equation: analysis of boundary layers, Asymptotic Analysis, 4 (1991), 293-317.  Google Scholar [15] Y. Sone, Kinetic Theory and Fluid Dynamics, Birkhauser Boston, 2002. Google Scholar [16] Y. Sone, Molecular Gas Dynamics, Theory, techniques, and applications. Modeling and Simulation in Science, Engineering and Technology. Birkh?user Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4573-1.  Google Scholar [17] S. Ukai, T. Yang and S.-H. Yu, Nonlinear boundary layers of the Boltzmann equation: I. Existence, Commun. Math. Phys., 236 (2003), 373-393. doi: 10.1007/s00220-003-0822-8.  Google Scholar [18] S. Ukai, T. Yang and S.-H. Yu, Nonlinear stability of boundary layers of the Boltzmann equation; I. The case $\mathcalM_\infty <-1$, Commun. Math. Phys., 244 (2004), 99-109. doi: 10.1007/s00220-003-0976-4.  Google Scholar
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