Revista de la
Unión Matemática Argentina
Coupling local and nonlocal equations with Neumann boundary conditions
Gabriel Acosta, Francisco Bersetche, and Julio D. Rossi

Volume 65, no. 2 (2023), pp. 533–565    

Published online: December 28, 2023

https://doi.org/10.33044/revuma.3046

Download PDF

Abstract

We introduce two different ways of coupling local and nonlocal equations with Neumann boundary conditions in such a way that the resulting model is naturally associated with an energy functional. For these two models we prove that there is a minimizer of the resulting energy that is unique modulo adding a constant.

References

  1. G. Acosta, F. Bersetche, and J. D. Rossi, Local and nonlocal energy-based coupling models, SIAM J. Math. Anal. 54 no. 6 (2022), 6288–6322.  DOI  MR  Zbl
  2. F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi, and J. J. Toledo-Melero, Nonlocal Diffusion Problems, {Mathematical Surveys and Monographs} {165}, American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010.  DOI  MR  Zbl
  3. Y. Azdoud, F. Han, and G. Lubineau, A morphing framework to couple non-local and local anisotropic continua, Int. J. Solids Struct. 50 no. 9 (2013), 1332–1341.  DOI
  4. S. Badia, P. Bochev, R. Lehoucq, M. Parks, J. Fish, M. A. Nuggehally, and M. Gunzburger, A force-based blending model for atomistic-to-continuum coupling, Int. J. Multiscale Comput. Eng. 5 no. 5 (2007), 387–406.  DOI
  5. S. Badia, M. Parks, P. Bochev, M. Gunzburger, and R. Lehoucq, On atomistic-to-continuum coupling by blending, Multiscale Model. Simul. 7 no. 1 (2008), 381–406.  DOI  MR  Zbl
  6. P. W. Bates and A. Chmaj, An integrodifferential model for phase transitions: stationary solutions in higher space dimensions, J. Statist. Phys. 95 no. 5-6 (1999), 1119–1139.  DOI  MR  Zbl
  7. H. Berestycki, A.-C. Coulon, J.-M. Roquejoffre, and L. Rossi, The effect of a line with nonlocal diffusion on {F}isher-{KPP} propagation, Math. Models Methods Appl. Sci. 25 no. 13 (2015), 2519–2562.  DOI  MR  Zbl
  8. H. Brezis, \emph{Functional Analysis, {S}obolev Spaces and Partial Differential Equations}, {Universitext}, Springer, New York, 2011.  MR  Zbl
  9. M. Capanna and J. D. Rossi, Mixing local and nonlocal evolution equations, Mediterr. J. Math. 20 no. 2 (2023), Paper No. 59, 36 pp.  DOI  MR  Zbl
  10. C. Carrillo and P. Fife, Spatial effects in discrete generation population models, J. Math. Biol. 50 no. 2 (2005), 161–188.  DOI  MR  Zbl
  11. E. Chasseigne, M. Chaves, and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl. (9) 86 no. 3 (2006), 271–291.  DOI  MR  Zbl
  12. C. Cortazar, M. Elgueta, J. D. Rossi, and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differential Equations 234 no. 2 (2007), 360–390.  DOI  MR  Zbl
  13. C. Cortazar, M. Elgueta, J. D. Rossi, and N. Wolanski, How to approximate the heat equation with {N}eumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal. 187 no. 1 (2008), 137–156.  DOI  MR  Zbl
  14. M. D'Elia and P. Bochev, Formulation, analysis and computation of an optimization-based local-to-nonlocal coupling method, Results Appl. Math. 9 (2021), Paper No. 100129, 20 pp.  DOI  MR  Zbl
  15. M. D'Elia, X. Li, P. Seleson, X. Tian, and Y. Yu, A review of local-to-nonlocal coupling methods in nonlocal diffusion and nonlocal mechanics, J. Peridyn. Nonlocal Model. 4 no. 1 (2022), 1–50.  DOI  MR
  16. 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 no. 11 (2016), 2218–2230.  DOI  MR  Zbl
  17. 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  MR  Zbl
  18. M. Di Paola, G. Failla, and M. Zingales, Physically-based approach to the mechanics of strong non-local linear elasticity theory, J. Elasticity 97 no. 2 (2009), 103–130.  DOI  MR  Zbl
  19. 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 no. 3 (2018), 1386–1404.  DOI  MR  Zbl
  20. L. C. Evans, Partial Differential Equations, second ed., {Graduate Studies in Mathematics} {19}, American Mathematical Society, Providence, RI, 2010.  MR  Zbl
  21. P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, 2003, pp. 153–191.  MR  Zbl
  22. C. G. Gal and M. Warma, Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces, Comm. Partial Differential Equations 42 no. 4 (2017), 579–625.  DOI  MR  Zbl
  23. A. Gárriz, F. Quirós, and J. D. Rossi, Coupling local and nonlocal evolution equations, Calc. Var. Partial Differential Equations 59 no. 4 (2020), Paper No. 112, 24 pp.  DOI  MR  Zbl
  24. F. Han and G. Lubineau, Coupling of nonlocal and local continuum models by the {A}rlequin approach, Internat. J. Numer. Methods Engrg. 89 no. 6 (2012), 671–685.  DOI  MR  Zbl
  25. V. Hutson, S. Martinez, K. Mischaikow, and G. T. Vickers, The evolution of dispersal, J. Math. Biol. 47 no. 6 (2003), 483–517.  DOI  MR  Zbl
  26. D. Kriventsov, Regularity for a local-nonlocal transmission problem, Arch. Ration. Mech. Anal. 217 no. 3 (2015), 1103–1195.  DOI  MR  Zbl
  27. T. Mengesha and Q. Du, The bond-based peridynamic system with {D}irichlet-type volume constraint, Proc. Roy. Soc. Edinburgh Sect. A 144 no. 1 (2014), 161–186.  DOI  MR  Zbl
  28. L. E. Payne and H. F. Weinberger, An optimal {P}oincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), 286–292 (1960).  DOI  MR  Zbl
  29. B. C. dos Santos, S. M. Oliva, and J. D. Rossi, A local/nonlocal diffusion model, Appl. Anal. 101 no. 15 (2022), 5213–5246.  DOI  MR  Zbl
  30. P. Seleson, S. Beneddine, and S. Prudhomme, A force-based coupling scheme for peridynamics and classical elasticity, Comput. Mater. Sci. 66 (2013), 34–49.  DOI
  31. P. Seleson and M. Gunzburger, Bridging methods for atomistic-to-continuum coupling and their implementation, Commun. Comput. Phys. 7 no. 4 (2010), 831–876.  DOI  MR  Zbl
  32. P. Seleson, M. Gunzburger, and M. L. Parks, Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains, Comput. Methods Appl. Mech. Engrg. 266 (2013), 185–204.  DOI  MR  Zbl
  33. S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids 48 no. 1 (2000), 175–209.  DOI  MR  Zbl
  34. S. A. Silling, M. Epton, O. Weckner, J. Xu, and E. Askari, Peridynamic states and constitutive modeling, J. Elasticity 88 no. 2 (2007), 151–184.  DOI  MR  Zbl
  35. S. A. Silling and R. B. Lehoucq, Peridynamic theory of solid mechanics, Adv. Appl. Mech. 44 no. 1 (2010), 73–168.  DOI
  36. C. Strickland, G. Dangelmayr, and P. D. Shipman, Modeling the presence probability of invasive plant species with nonlocal dispersal, J. Math. Biol. 69 no. 2 (2014), 267–294.  DOI  MR  Zbl
  37. X. Wang, Metastability and stability of patterns in a convolution model for phase transitions, J. Differential Equations 183 no. 2 (2002), 434–461.  DOI  MR  Zbl
  38. L. Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks, J. Differential Equations 197 no. 1 (2004), 162–196.  DOI  MR  Zbl