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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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