Homogenization of the heat equation with random convolutional potential

Mengmeng Wang; Dong Su; Wei Wang; Mengmeng Wang; Dong Su; Wei Wang

doi:10.3934/math.2024273

AIMS Mathematics

2024, Volume 9, Issue 3: 5661-5670. doi: 10.3934/math.2024273

Previous Article Next Article

Research article

Homogenization of the heat equation with random convolutional potential

Department of Mathematics, Nanjing University, Nanjing 210093, China

Received: 16 October 2023 Revised: 28 December 2023 Accepted: 09 January 2024 Published: 30 January 2024
MSC : 35B27, 35K05, 60J45

This paper derived the homogenization of the heat equation with random convolutional potential. By Tartar's method of oscillating test function, the solution of the heat equation with random convolutional potential was shown to converge in distribution to the solution of the effective equation with determined convolutional potential.
- homogenization,
- weak convergence,
- random convolutional potential,
- heat equation
Citation: Mengmeng Wang, Dong Su, Wei Wang. Homogenization of the heat equation with random convolutional potential[J]. AIMS Mathematics, 2024, 9(3): 5661-5670. doi: 10.3934/math.2024273

Related Papers:

Abstract

This paper derived the homogenization of the heat equation with random convolutional potential. By Tartar's method of oscillating test function, the solution of the heat equation with random convolutional potential was shown to converge in distribution to the solution of the effective equation with determined convolutional potential.

References

[1]	G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482–1518. https://doi.org/10.1137/0523084 doi: 10.1137/0523084
[2]	B. Amaziane, L. Pankratov, A. Piatnitski, Homogenization of immiscible compressible two-phase flow in random porous media, J. Differ. Equations, 305 (2021), 206–223. https://doi.org/10.1016/j.jde.2021.10.012 doi: 10.1016/j.jde.2021.10.012
[3]	G. Bal, Convergence to SPDEs in Stratonovich form, Commun. Math. Phys., 292 (2009), 457–477. https://doi.org/10.1007/s00220-009-0898-x doi: 10.1007/s00220-009-0898-x
[4]	G. Bal, Convergence to homogenized or stochastic partial differential equations, Appl. Math. Res. Express, 2011 (2011), 215–241. https://doi.org/10.1093/amrx/abr006 doi: 10.1093/amrx/abr006
[5]	A. Bensoussan, J. L. Lions, G. Papanicolaou, Asymptotic analysis for periodic structures, AMS Chelsea Publishing, 1978. https://doi.org/10.1090/chel/374
[6]	H. Bessaih, Y. Efendiev, R. F. Maris, Stochastic homogenization for a diffusion-reaction model, Discrete Cont. Dyn. Syst., 39 (2019), 5403–5429. https://doi.org/10.3934/dcds.2019221 doi: 10.3934/dcds.2019221
[7]	H. Bessaih, Y. Efendiev, R. F. Maris, Stochastic homogenization of a convection-diffusion equation, SIAM J. Math. Anal., 53 (2021), 2718–2745. https://doi.org/10.1137/19M1302776 doi: 10.1137/19M1302776
[8]	P. Billingsley, Convergence of probability measures, John Wiley & Sons, 1999. https://doi.org/10.1002/9780470316962
[9]	G. A. Chechkin, A. L. Piatnitski, A. S. Shamaev, Homogenization: Methods and applications, American Mathematical Society, 2007. https://doi.org/10.1090/mmono/234
[10]	D. Cioranescu, P. Donato, An Introduction to homogenization, Oxford University Press, 1999. https://doi.org/10.1093/oso/9780198565543.001.0001
[11]	F. Golse, On the dynamics of large particle systems in the mean field limit, In: Macroscopic and large scale phenomena: Coarse graining, mean field limits and ergodicity, Lecture Notes in Applied Mathematics and Mechanics, Springer, Cham, 2016. https://doi.org/10.1007/978-3-319-26883-5_1
[12]	Y. Gu, G. Bal, Homogenization of parabolic equations with large time-dependent random potential, Stoch. Proc. Appl., 125 (2015), 91–115. https://doi.org/10.1016/j.spa.2014.07.024 doi: 10.1016/j.spa.2014.07.024
[13]	S. Gu, J. P. Zhuge, Periodic homogenization of Green's functions for stokes systems, Stoch. Proc. Appl., 58 (2019), 1–46. https://doi.org/10.1007/s00526-019-1553-9 doi: 10.1007/s00526-019-1553-9
[14]	M. Hairer, E. Pardoux, A. Piatnitski, Random homogenization of a highly oscillatory singular potential, Stoch. Partial Differ., 1 (2013), 571–605. https://doi.org/10.1007/s40072-013-0018-y doi: 10.1007/s40072-013-0018-y
[15]	B. Iftimie, É. Pardoux, A. Piatnitski, Homogenization of a singular random one-dimensional PDE, Ann. Inst. H. Poincaré Probab. Statist., 44 (2008), 519–543. https://doi.org/10.1214/07-AIHP134 doi: 10.1214/07-AIHP134
[16]	V. V. Jikov, S. M. Kozlov, O. A. Oleinik, Homogenization of differential operators and integral functionals, Heidelberg: Springer-Verlag Berlin, 1994. https://doi.org/10.1007/978-3-642-84659-5
[17]	T. Komorowski, E. Nieznaj, On the asymptotic behavior of solutions of the heat equation with a random, long-range correlated potential, Potential Anal., 33 (2010), 175–197. https://doi.org/10.1007/s11118-009-9164-2 doi: 10.1007/s11118-009-9164-2
[18]	M. S. Kozlov, Averaging of random operators, Sb. Math., 37 (1980), 167–180. https://doi.org/10.1070/SM1980v037n02ABEH001948 doi: 10.1070/SM1980v037n02ABEH001948
[19]	J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Pairs: Dunod, 1969.
[20]	M. Mohammed, M. Sango, Homogenization of linear hyperbolic stochastic partial differential equation with rapidly oscillating coefficients: The two scale convergence method, Asymptot. Anal., 91 (2015), 341–371. https://doi.org/10.3233/ASY-141269 doi: 10.3233/ASY-141269
[21]	M. Mohammed, Homogenization of nonlinear hyperbolic stochastic equation via Tartar's method, J. Hyperbol. Differ. Eq., 14 (2017), 323–340. https://doi.org/10.1142/S0219891617500096 doi: 10.1142/S0219891617500096
[22]	É. Pardoux, A. Piatnitski, Homogenization of a singular random one-dimensional PDE with time-varying coefficients, Ann. Probab., 40 (2012), 1316–1356. https://doi.org/10.1214/11-AOP650 doi: 10.1214/11-AOP650
[23]	Z. W. Shen, Periodic homogenization of elliptic systems, Cham: Birkhäuser, 2018. https://doi.org/10.1007/978-3-319-91214-1
[24]	L. Tartar, The general theory of homogenization: A personalized introduction, Heidelberg: Springer-Verlag Berlin, 2009.
[25]	W. Wang, D. M. Cao, J. Q. Duan, Effective macroscopic dynamics of stochastic partial equations in perforated domains, SIAM J. Math. Anal., 38 (2007), 1508–1527. https://doi.org/10.1137/050648766 doi: 10.1137/050648766
[26]	W. Wang, J. Q. Duan, Homogenized dynamics of stochastic partial differential equation with dynamical boundary conditions, Commun. Math. Phys., 275 (2007), 163–186. https://doi.org/10.1007/s00220-007-0301-8 doi: 10.1007/s00220-007-0301-8
[27]	N. Y. Zhang, G. Bal, Convergence to SPDE of the Schrödinger equation with large, random potential, Commun. Math. Sci., 12 (2014), 825–841. https://dx.doi.org/10.4310/CMS.2014.v12.n5.a2 doi: 10.4310/CMS.2014.v12.n5.a2
[28]	V. V. Zhikov, A. L. Piatnitski, Homogenization of random singular structures and random measures, Izv. Math., 70 (2006), 19–67. https://dx.doi.org/10.1070/IM2006v070n01ABEH002302 doi: 10.1070/IM2006v070n01ABEH002302

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)