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

Diffusion limit for a kinetic equation with a thermostatted interface

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 $.

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. BardosF. 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. BardosE. BernardF. 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. BardosR. 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. BasileS. 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 AbdallahA. 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. BensoussanJ. 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. JaraT. 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. MelletS. 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. BardosF. 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. BardosE. BernardF. 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. BardosR. 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. BasileS. 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 AbdallahA. 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. BensoussanJ. 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. JaraT. 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. MelletS. 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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