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Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics
Homogenization for nonlocal problems with smooth kernels
1. | CONICET and Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Ciudad Universitaria, Pabellon I, (1428), Buenos Aires, Argentina |
2. | Dpto. de Matemática, ICMC, Universidade de São Paulo, Avenida Trabalhador São-Carlense, 400, São Carlos - SP, Brazil |
3. | Dpto. de Matemática Aplicada, IME, Universidade de São Paulo, Rua do Matão 1010, São Paulo - SP, Brazil |
In this paper we consider the homogenization problem for a nonlocal equation that involve different smooth kernels. We assume that the spacial domain is divided into a sequence of two subdomains $ A_n \cup B_n $ and we have three different smooth kernels, one that controls the jumps from $ A_n $ to $ A_n $, a second one that controls the jumps from $ B_n $ to $ B_n $ and the third one that governs the interactions between $ A_n $ and $ B_n $. Assuming that $ \chi_{A_n} (x) \to X(x) $ weakly-* in $ L^\infty $ (and then $ \chi_{B_n} (x) \to (1-X)(x) $ weakly-* in $ L^\infty $) as $ n \to \infty $ we show that there is an homogenized limit system in which the three kernels and the limit function $ X $ appear. We deal with both Neumann and Dirichlet boundary conditions. Moreover, we also provide a probabilistic interpretation of our results.
References:
[1] |
F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, vol. 165. AMS, 2010.
doi: 10.1090/surv/165. |
[2] |
P. W. Bates and A. Chmaj,
An integrodifferential model for phase transitions: Stationary solutions in higher dimensions, J. Statist. Phys., 95 (1999), 1119-1139.
doi: 10.1023/A:1004514803625. |
[3] |
A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland Publishing Company, 1978. |
[4] |
L. Caffarelli and A. Mellet,
Random homogenization of fractional obstacle problems, Netw. Heterog. Media, 3 (2008), 523-554.
doi: 10.3934/nhm.2008.3.523. |
[5] |
M. Capanna and J. D. Rossi, Mixing local and nonlocal evolution equations, Preprint, arXiv: 2003.03407v1. Google Scholar |
[6] |
C. Carrillo and P. Fife,
Spatial effects in discrete generation population models., J. Math. Biol., 50 (2005), 161-188.
doi: 10.1007/s00285-004-0284-4. |
[7] |
P. Cazeaux and C. Grandmont,
Homogenization of a multiscale viscoelastic model with nonlocal damping, application to the human lungs, Math. Models Methods Appl. Sci., 25 (2015), 1125-1177.
doi: 10.1142/S0218202515500293. |
[8] |
E. Chasseigne, M. Chaves and J. D. Rossi,
Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291.
doi: 10.1016/j.matpur.2006.04.005. |
[9] |
D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, New York, 1999.
![]() |
[10] |
C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski,
How to approximate the heat equation with neumann boundary conditions by nonlocal diffusion problems, Arch. Rational Mech. Anal., 187 (2008), 137-156.
doi: 10.1007/s00205-007-0062-8. |
[11] |
M. D'Elia, Q. Du, M. Gunzburger and R. Lehoucq,
Nonlocal convection-diffusion problems on bounded domains and finite-range jump processes, Comput. Methods Appl. Math., 17 (2017), 707-722.
doi: 10.1515/cmam-2017-0029. |
[12] |
M. D'Elia, M. Perego, P. Bochev and D. Littlewood,
A coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions, Comput. Math. Appl., 71 (2016), 2218-2230.
doi: 10.1016/j.camwa.2015.12.006. |
[13] |
M. D'Elia, D. Ridzal, K. J. Peterson, P. Bochev and M. Shashkov,
Optimization-based mesh correction with volume and convexity constraints, J. Comput. Phys., 313 (2016), 455-477.
doi: 10.1016/j.jcp.2016.02.050. |
[14] |
Q. Du, X. H. Li, J. Lu and X. Tian,
A quasi-nonlocal coupling method for nonlocal and local diffusion models, SIAM J. Numer. Anal., 56 (2018), 1386-1404.
doi: 10.1137/17M1124012. |
[15] |
P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, In Trends in Nonlinear Analysis, 153-191, Springer, Berlin, 2003. |
[16] |
C. G. Gal and M. Warma,
Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces, Comm. Partial Differential Equations, 42 (2017), 579-625.
doi: 10.1080/03605302.2017.1295060. |
[17] |
A. Gárriz, F. Quirós and J. D. Rossi, Coupling local and nonlocal evolution equations, Calc. Var. Par. Diff. Equations., 59 (2020), Paper No. 112, 24 pp. arXiv: 1903.07108.
doi: 10.1007/s00526-020-01771-z. |
[18] |
C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems, Grundlehren der mathematischen Wissenschaften, Springer, Berlin, New York, 1999.
doi: 10.1007/978-3-662-03752-2. |
[19] |
D. Kriventsov,
Regularity for a local-nonlocal transmission problem, Arch. Ration. Mech. Anal., 217 (2015), 1103-1195.
doi: 10.1007/s00205-015-0851-4. |
[20] |
T. M. Liggett, Interacting Particle Systems, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, 1985.
doi: 10.1007/978-1-4613-8542-4. |
[21] |
M. C. Pereira,
Nonlocal evolution equations in perforated domains, Math. Methods Appl. Sciences, 41 (2018), 6368-6377.
doi: 10.1002/mma.5144. |
[22] |
M. C. Pereira and J. D. Rossi,
An obstacle problem for nonlocal equations in perforated domains, Potential Analysis, 48 (2018), 361-373.
doi: 10.1007/s11118-017-9639-5. |
[23] |
M. C. Pereira and J. D. Rossi,
Nonlocal problems in perforated domains, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 305-340.
doi: 10.1017/prm.2018.130. |
[24] |
R. W. Schwab,
Periodic homogenization for nonlinear integro-differential equations, SIAM J. Math. Anal., 42 (2010), 2652-2680.
doi: 10.1137/080737897. |
[25] |
L. Tartar, The General Theory of Homogenization. A Personalized Introduction, Lecture Notes of the Unione Matematica Italiana, Springer-Verlag, 2009.
doi: 10.1007/978-3-642-05195-1. |
[26] |
V. S. Varadarajan,
Weak convergence of measures on separable metric spaces, The Indian Journal of Statistics., 19 (1958), 15-22.
|
[27] |
M. Waurick,
Homogenization in fractional elasticity, SIAM J. Math. Anal., 46 (2014), 1551-1576.
doi: 10.1137/130941596. |
[28] |
D. Williams, Probability with Martingales, Cambridge University Press, 1991.
doi: 10.1017/CBO9780511813658.![]() ![]() |
show all references
References:
[1] |
F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, vol. 165. AMS, 2010.
doi: 10.1090/surv/165. |
[2] |
P. W. Bates and A. Chmaj,
An integrodifferential model for phase transitions: Stationary solutions in higher dimensions, J. Statist. Phys., 95 (1999), 1119-1139.
doi: 10.1023/A:1004514803625. |
[3] |
A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland Publishing Company, 1978. |
[4] |
L. Caffarelli and A. Mellet,
Random homogenization of fractional obstacle problems, Netw. Heterog. Media, 3 (2008), 523-554.
doi: 10.3934/nhm.2008.3.523. |
[5] |
M. Capanna and J. D. Rossi, Mixing local and nonlocal evolution equations, Preprint, arXiv: 2003.03407v1. Google Scholar |
[6] |
C. Carrillo and P. Fife,
Spatial effects in discrete generation population models., J. Math. Biol., 50 (2005), 161-188.
doi: 10.1007/s00285-004-0284-4. |
[7] |
P. Cazeaux and C. Grandmont,
Homogenization of a multiscale viscoelastic model with nonlocal damping, application to the human lungs, Math. Models Methods Appl. Sci., 25 (2015), 1125-1177.
doi: 10.1142/S0218202515500293. |
[8] |
E. Chasseigne, M. Chaves and J. D. Rossi,
Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291.
doi: 10.1016/j.matpur.2006.04.005. |
[9] |
D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, New York, 1999.
![]() |
[10] |
C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski,
How to approximate the heat equation with neumann boundary conditions by nonlocal diffusion problems, Arch. Rational Mech. Anal., 187 (2008), 137-156.
doi: 10.1007/s00205-007-0062-8. |
[11] |
M. D'Elia, Q. Du, M. Gunzburger and R. Lehoucq,
Nonlocal convection-diffusion problems on bounded domains and finite-range jump processes, Comput. Methods Appl. Math., 17 (2017), 707-722.
doi: 10.1515/cmam-2017-0029. |
[12] |
M. D'Elia, M. Perego, P. Bochev and D. Littlewood,
A coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions, Comput. Math. Appl., 71 (2016), 2218-2230.
doi: 10.1016/j.camwa.2015.12.006. |
[13] |
M. D'Elia, D. Ridzal, K. J. Peterson, P. Bochev and M. Shashkov,
Optimization-based mesh correction with volume and convexity constraints, J. Comput. Phys., 313 (2016), 455-477.
doi: 10.1016/j.jcp.2016.02.050. |
[14] |
Q. Du, X. H. Li, J. Lu and X. Tian,
A quasi-nonlocal coupling method for nonlocal and local diffusion models, SIAM J. Numer. Anal., 56 (2018), 1386-1404.
doi: 10.1137/17M1124012. |
[15] |
P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, In Trends in Nonlinear Analysis, 153-191, Springer, Berlin, 2003. |
[16] |
C. G. Gal and M. Warma,
Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces, Comm. Partial Differential Equations, 42 (2017), 579-625.
doi: 10.1080/03605302.2017.1295060. |
[17] |
A. Gárriz, F. Quirós and J. D. Rossi, Coupling local and nonlocal evolution equations, Calc. Var. Par. Diff. Equations., 59 (2020), Paper No. 112, 24 pp. arXiv: 1903.07108.
doi: 10.1007/s00526-020-01771-z. |
[18] |
C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems, Grundlehren der mathematischen Wissenschaften, Springer, Berlin, New York, 1999.
doi: 10.1007/978-3-662-03752-2. |
[19] |
D. Kriventsov,
Regularity for a local-nonlocal transmission problem, Arch. Ration. Mech. Anal., 217 (2015), 1103-1195.
doi: 10.1007/s00205-015-0851-4. |
[20] |
T. M. Liggett, Interacting Particle Systems, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, 1985.
doi: 10.1007/978-1-4613-8542-4. |
[21] |
M. C. Pereira,
Nonlocal evolution equations in perforated domains, Math. Methods Appl. Sciences, 41 (2018), 6368-6377.
doi: 10.1002/mma.5144. |
[22] |
M. C. Pereira and J. D. Rossi,
An obstacle problem for nonlocal equations in perforated domains, Potential Analysis, 48 (2018), 361-373.
doi: 10.1007/s11118-017-9639-5. |
[23] |
M. C. Pereira and J. D. Rossi,
Nonlocal problems in perforated domains, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 305-340.
doi: 10.1017/prm.2018.130. |
[24] |
R. W. Schwab,
Periodic homogenization for nonlinear integro-differential equations, SIAM J. Math. Anal., 42 (2010), 2652-2680.
doi: 10.1137/080737897. |
[25] |
L. Tartar, The General Theory of Homogenization. A Personalized Introduction, Lecture Notes of the Unione Matematica Italiana, Springer-Verlag, 2009.
doi: 10.1007/978-3-642-05195-1. |
[26] |
V. S. Varadarajan,
Weak convergence of measures on separable metric spaces, The Indian Journal of Statistics., 19 (1958), 15-22.
|
[27] |
M. Waurick,
Homogenization in fractional elasticity, SIAM J. Math. Anal., 46 (2014), 1551-1576.
doi: 10.1137/130941596. |
[28] |
D. Williams, Probability with Martingales, Cambridge University Press, 1991.
doi: 10.1017/CBO9780511813658.![]() ![]() |
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