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September  2015, 8(3): 467-492. doi: 10.3934/krm.2015.8.467

Diffusion limit for the radiative transfer equation perturbed by a Wiener process

1. 

IRMAR, ENS Rennes, CNRS, UEB, av. Robert Schuman, F-35170 Bruz, France

2. 

IRMAR, ENS Rennes, av. Robert Schuman, F-35170 Bruz, France

3. 

Université de Lyon, CNRS UMR 5208 & Université Lyon 1, Institut Camille Jordan, 43 bd du 11 novembre 1918, F-69622 Villeurbanne cedex

Received  October 2014 Revised  April 2015 Published  June 2015

The aim of this paper is the rigorous derivation of a stochastic non-linear diffusion equation from a radiative transfer equation perturbed with a random noise. The proof of the convergence relies on a formal Hilbert expansion and the estimation of the remainder. The Hilbert expansion has to be done up to order 3 to overcome some difficulties caused by the random noise.
Citation: Arnaud Debussche, Sylvain De Moor, Julien Vovelle. Diffusion limit for the radiative transfer equation perturbed by a Wiener process. Kinetic & Related Models, 2015, 8 (3) : 467-492. doi: 10.3934/krm.2015.8.467
References:
[1]

C. Bardos, F. Golse and B. Perthame, The Rosseland approximation for the radiative transfer equations, Comm. Pure Appl. Math., 40 (1987), 691-721. doi: 10.1002/cpa.3160400603.  Google Scholar

[2]

C. Bardos, F. Golse, B. Perthame and R. Sentis, The nonaccretive radiative transfer equations: Existence of solutions and Rosseland approximation, J. Funct. Anal., 77 (1988), 434-460. doi: 10.1016/0022-1236(88)90096-1.  Google Scholar

[3]

P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968.  Google Scholar

[4]

P. Bouchut and L. Desvillettes, Averaging lemmas without time Fourier transform and application to discretized kinetic equations, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 19-36. doi: 10.1017/S030821050002744X.  Google Scholar

[5]

Z. Brzeźniak, On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep., 61 (1997), 245-295. doi: 10.1080/17442509708834122.  Google Scholar

[6]

Z. Brzeźniak and S. Peszat, Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process, Studia Math., 137 (1999), 261-299.  Google Scholar

[7]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar

[8]

A. de Bouard and A. Debussche, A stochastic nonlinear Schrödinger equation with multiplicative noise, Comm. Math. Phys., 205 (1999), 161-181. doi: 10.1007/s002200050672.  Google Scholar

[9]

A. de Bouard and M. Gazeau, A diffusion approximation theorem for a nonlinear PDE with application to random birefringent optical fibers, Ann. Appl. Probab., 22 (2012), 2460-2504. doi: 10.1214/11-AAP839.  Google Scholar

[10]

A. Debussche and J. Vovelle, Diffusion limit for a stochastic kinetic problem, Commun. Pure Appl. Anal., 11 (2012), 2305-2326. doi: 10.3934/cpaa.2012.11.2305.  Google Scholar

[11]

A. Debussche, S. De Moor and M. Hofmanová, A regularity result for quasilinear stochastic partial differential equations of parabolic type, SIAM Journal on Mathematical Analysis, 47 (2015), 1590-1614. doi: 10.1137/130950549.  Google Scholar

[12]

A. Debussche, S. De Moor and J. Vovelle, Diffusion limit for the radiative transfer equation perturbed by a Markovian process,, preprint, ().   Google Scholar

[13]

J. P. Fouque, J. Garnier, G. Papanicolaou and K. Solna, Wave Propagation and Time Reversal in Randomly Layered Media, Stochastic Modelling and Applied Probability, 56, Springer, New York, 2007. doi: 10.1007/978-0-387-49808-9_4.  Google Scholar

[14]

I. Gyöngy, Existence and uniqueness results for semilinear stochastic partial differential equations, Stochastic Process. Appl., 73 (1998), 271-299. doi: 10.1016/S0304-4149(97)00103-8.  Google Scholar

[15]

I. Gyöngy and N. Krylov, Existence of strong solutions for Itô's stochastic equations via approximations, Probab. Theory Related Fields, 105 (1996), 143-158. doi: 10.1007/BF01203833.  Google Scholar

[16]

P.-L. Lions, B. Perthame and P. E. Souganidis, Stochastic averaging lemmas for kinetic equations,, in S'eminaire Laurent Schwartz - EDP et applications (2011-2012), (): 2011.  doi: 10.5802/slsedp.21.  Google Scholar

[17]

A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations, Comm. Math. Phys., 281 (2008), 573-596. doi: 10.1007/s00220-008-0523-4.  Google Scholar

[18]

G. C. Papanicolaou, D. Stroock and S. R. S. Varadhan, Martingale approach to some limit theorems, in Papers from the Duke Turbulence Conference (Duke Univ., Durham, N.C., 1976), Academic Press, 1977.  Google Scholar

[19]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2007. doi: 10.1017/CBO9780511721373.  Google Scholar

[20]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

show all references

References:
[1]

C. Bardos, F. Golse and B. Perthame, The Rosseland approximation for the radiative transfer equations, Comm. Pure Appl. Math., 40 (1987), 691-721. doi: 10.1002/cpa.3160400603.  Google Scholar

[2]

C. Bardos, F. Golse, B. Perthame and R. Sentis, The nonaccretive radiative transfer equations: Existence of solutions and Rosseland approximation, J. Funct. Anal., 77 (1988), 434-460. doi: 10.1016/0022-1236(88)90096-1.  Google Scholar

[3]

P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968.  Google Scholar

[4]

P. Bouchut and L. Desvillettes, Averaging lemmas without time Fourier transform and application to discretized kinetic equations, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 19-36. doi: 10.1017/S030821050002744X.  Google Scholar

[5]

Z. Brzeźniak, On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep., 61 (1997), 245-295. doi: 10.1080/17442509708834122.  Google Scholar

[6]

Z. Brzeźniak and S. Peszat, Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process, Studia Math., 137 (1999), 261-299.  Google Scholar

[7]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar

[8]

A. de Bouard and A. Debussche, A stochastic nonlinear Schrödinger equation with multiplicative noise, Comm. Math. Phys., 205 (1999), 161-181. doi: 10.1007/s002200050672.  Google Scholar

[9]

A. de Bouard and M. Gazeau, A diffusion approximation theorem for a nonlinear PDE with application to random birefringent optical fibers, Ann. Appl. Probab., 22 (2012), 2460-2504. doi: 10.1214/11-AAP839.  Google Scholar

[10]

A. Debussche and J. Vovelle, Diffusion limit for a stochastic kinetic problem, Commun. Pure Appl. Anal., 11 (2012), 2305-2326. doi: 10.3934/cpaa.2012.11.2305.  Google Scholar

[11]

A. Debussche, S. De Moor and M. Hofmanová, A regularity result for quasilinear stochastic partial differential equations of parabolic type, SIAM Journal on Mathematical Analysis, 47 (2015), 1590-1614. doi: 10.1137/130950549.  Google Scholar

[12]

A. Debussche, S. De Moor and J. Vovelle, Diffusion limit for the radiative transfer equation perturbed by a Markovian process,, preprint, ().   Google Scholar

[13]

J. P. Fouque, J. Garnier, G. Papanicolaou and K. Solna, Wave Propagation and Time Reversal in Randomly Layered Media, Stochastic Modelling and Applied Probability, 56, Springer, New York, 2007. doi: 10.1007/978-0-387-49808-9_4.  Google Scholar

[14]

I. Gyöngy, Existence and uniqueness results for semilinear stochastic partial differential equations, Stochastic Process. Appl., 73 (1998), 271-299. doi: 10.1016/S0304-4149(97)00103-8.  Google Scholar

[15]

I. Gyöngy and N. Krylov, Existence of strong solutions for Itô's stochastic equations via approximations, Probab. Theory Related Fields, 105 (1996), 143-158. doi: 10.1007/BF01203833.  Google Scholar

[16]

P.-L. Lions, B. Perthame and P. E. Souganidis, Stochastic averaging lemmas for kinetic equations,, in S'eminaire Laurent Schwartz - EDP et applications (2011-2012), (): 2011.  doi: 10.5802/slsedp.21.  Google Scholar

[17]

A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations, Comm. Math. Phys., 281 (2008), 573-596. doi: 10.1007/s00220-008-0523-4.  Google Scholar

[18]

G. C. Papanicolaou, D. Stroock and S. R. S. Varadhan, Martingale approach to some limit theorems, in Papers from the Duke Turbulence Conference (Duke Univ., Durham, N.C., 1976), Academic Press, 1977.  Google Scholar

[19]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2007. doi: 10.1017/CBO9780511721373.  Google Scholar

[20]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

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