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October  2019, 12(5): 1185-1196. doi: 10.3934/krm.2019045

Diffusion limit for a kinetic equation with a thermostatted interface

Giada Basile 1, , Tomasz Komorowski 2, and  Stefano Olla 3,,

1. 

Dipartimento di Matematica, Università di Roma La Sapienza, Roma, Italy
2. 

IMPAN, Polish Academy of Sciences, Warsaw, Poland
3. 

CEREMADE, UMR CNRS, Université Paris-Dauphine, PSL Research University, 75016 Paris, France

* Corresponding author: Stefano Olla

Received  March 2019 Published  July 2019

Fund Project: T.K. acknowledges the support of the National Science Centre: NCN grant 2016/23/B/ST1/00492. SO's research is supported by ANR-15-CE40-0020-01 grant LSD. Both T.K. and S.O. were partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.

We consider a linear phonon Boltzmann equation with a reflecting/transmitting/absorbing interface. This equation appears as the Boltzmann-Grad limit for the energy density function of a harmonic chain of oscillators with inter-particle stochastic scattering in the presence of a heat bath at temperature $ T $ in contact with one oscillator at the origin. We prove that under the diffusive scaling the solutions of the phonon equation tend to the solution $ \rho(t, y) $ of a heat equation with the boundary condition $ \rho(t, 0)\equiv T $.

Keywords: Diffusion limit, thermostatted boundary conditions.
Mathematics Subject Classification: Primary: 60F17; Secondary: 82C70.
Citation: Giada Basile, Tomasz Komorowski, Stefano Olla. Diffusion limit for a kinetic equation with a thermostatted interface. Kinetic and Related Models, 2019, 12 (5) : 1185-1196. doi: 10.3934/krm.2019045
References:
[1]

G. Bal and L. Ryzhik, Diffusion approximation of radiative transfer problems with interfaces, SIAM Journal on Applied Mathematics, 60 (2000), 1887-1912.  doi: 10.1137/S0036139999352080.

[2]

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.

[3]

C. Bardos, E. Bernard, F. Golse and R. Sentis, The diffusion approximation for the linear boltzmann equation with vanishing scattering coefficient, Communications in Math. Sciences, 13 (2015), 641-671.  doi: 10.4310/CMS.2015.v13.n3.a3.

[4]

C. Bardos, R. Santos and R. Sentis, Diffusion approximation and computation of the critical size, J Trans. Amer. Math. Soc., 284 (1984), 617-649.  doi: 10.1090/S0002-9947-1984-0743736-0.

[5]

G. Basile, From a kinetic equation to a diffusion under an anomalous scaling, Ann. Inst. H. Poincaré Probab. Statist., 50 (2014), 1301-1322.  doi: 10.1214/13-AIHP554.

[6]

G. Basile and A. Bovier, Convergence of a kinetic equation to a fractional diffusion equation, Markov Process. Related Fields, 16 (2010), 15-44. 

[7]

G. Basile, S. Olla and H. Spohn, Wigner functions and stochastically perturbed lattice dynamics, Archive for Rational Mechanics and Analysis, 195 (2009), 171-203.  doi: 10.1007/s00205-008-0205-6.

[8]

N. Ben Abdallah, A. Mellet and M. Puel, Fractional diffusion limit for collisional kinetic equations: a Hilbert expansion approach, Kinet. Relat. Models, 4 (2011), 873-900.  doi: 10.3934/krm.2011.4.873.

[9]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Boundary Layers and Homogenization of transport processes, Publ. RIMS, Kyoto Univ., 15 (1979), 53-157.  doi: 10.2977/prims/1195188427.

[10]

L. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, 6, Springer Verlag, Berlin, 1993. doi: 10.1007/978-3-642-58004-8.

[11]

M. Jara, T. Komorowski and S. Olla, Limit theorems for additive functionals of a Markov chain, Ann. of Appl. Prob., 19 (2009), 2270-2300.  doi: 10.1214/09-AAP610.

[12]

T. Komorowski, S. Olla and L. Ryzhik, Fractional diffusion limit for a kinetic equation with an interface - probabilistic approach, in preparation.

[13]

T. Komorowski, S. Olla, L. Ryzhik and H. Spohn, High frequency limit for a chain of harmonic oscillators with a point Langevin thermostat, 2018, preprint arXiv: 1806.02089

[14]

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

[15]

A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Archive for Rational Mechanics and Analysis, 199 (2011), 493-525.  doi: 10.1007/s00205-010-0354-2.

show all references

References:
[1]

G. Bal and L. Ryzhik, Diffusion approximation of radiative transfer problems with interfaces, SIAM Journal on Applied Mathematics, 60 (2000), 1887-1912.  doi: 10.1137/S0036139999352080.

[2]

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.

[3]

C. Bardos, E. Bernard, F. Golse and R. Sentis, The diffusion approximation for the linear boltzmann equation with vanishing scattering coefficient, Communications in Math. Sciences, 13 (2015), 641-671.  doi: 10.4310/CMS.2015.v13.n3.a3.

[4]

C. Bardos, R. Santos and R. Sentis, Diffusion approximation and computation of the critical size, J Trans. Amer. Math. Soc., 284 (1984), 617-649.  doi: 10.1090/S0002-9947-1984-0743736-0.

[5]

G. Basile, From a kinetic equation to a diffusion under an anomalous scaling, Ann. Inst. H. Poincaré Probab. Statist., 50 (2014), 1301-1322.  doi: 10.1214/13-AIHP554.

[6]

G. Basile and A. Bovier, Convergence of a kinetic equation to a fractional diffusion equation, Markov Process. Related Fields, 16 (2010), 15-44. 

[7]

G. Basile, S. Olla and H. Spohn, Wigner functions and stochastically perturbed lattice dynamics, Archive for Rational Mechanics and Analysis, 195 (2009), 171-203.  doi: 10.1007/s00205-008-0205-6.

[8]

N. Ben Abdallah, A. Mellet and M. Puel, Fractional diffusion limit for collisional kinetic equations: a Hilbert expansion approach, Kinet. Relat. Models, 4 (2011), 873-900.  doi: 10.3934/krm.2011.4.873.

[9]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Boundary Layers and Homogenization of transport processes, Publ. RIMS, Kyoto Univ., 15 (1979), 53-157.  doi: 10.2977/prims/1195188427.

[10]

L. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, 6, Springer Verlag, Berlin, 1993. doi: 10.1007/978-3-642-58004-8.

[11]

M. Jara, T. Komorowski and S. Olla, Limit theorems for additive functionals of a Markov chain, Ann. of Appl. Prob., 19 (2009), 2270-2300.  doi: 10.1214/09-AAP610.

[12]

T. Komorowski, S. Olla and L. Ryzhik, Fractional diffusion limit for a kinetic equation with an interface - probabilistic approach, in preparation.

[13]

T. Komorowski, S. Olla, L. Ryzhik and H. Spohn, High frequency limit for a chain of harmonic oscillators with a point Langevin thermostat, 2018, preprint arXiv: 1806.02089

[14]

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

[15]

A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Archive for Rational Mechanics and Analysis, 199 (2011), 493-525.  doi: 10.1007/s00205-010-0354-2.

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