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Uniform estimates on the Fisher information for solutions to Boltzmann and Landau equations
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 |
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 $.
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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