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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. 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. doi: 10.1016/0022-1236(88)90096-1. Google Scholar [3] P. Billingsley, Convergence of Probability Measures,, John Wiley & Sons, (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. doi: 10.1017/S030821050002744X. Google Scholar [5] Z. Brzeźniak, On stochastic convolution in Banach spaces and applications,, Stochastics Stochastics Rep., 61 (1997), 245. 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. Google Scholar [7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Encyclopedia of Mathematics and its Applications, (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. 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. 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. 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. 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, (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. 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. 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. 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., (1976). 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, (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. doi: 10.1007/BF01762360. Google Scholar

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References:
 [1] C. Bardos, F. Golse and B. Perthame, The Rosseland approximation for the radiative transfer equations,, Comm. Pure Appl. Math., 40 (1987), 691. 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. doi: 10.1016/0022-1236(88)90096-1. Google Scholar [3] P. Billingsley, Convergence of Probability Measures,, John Wiley & Sons, (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. doi: 10.1017/S030821050002744X. Google Scholar [5] Z. Brzeźniak, On stochastic convolution in Banach spaces and applications,, Stochastics Stochastics Rep., 61 (1997), 245. 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. Google Scholar [7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Encyclopedia of Mathematics and its Applications, (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. 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. 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. 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. 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, (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. 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. 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. 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., (1976). 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, (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. doi: 10.1007/BF01762360. Google Scholar
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