A new nonlocal calculus framework. Helmholtz decompositions, properties, and convergence for nonlocal operators in the limit of the vanishing horizon. (English) Zbl 07532101
Summary: We introduce a new nonlocal calculus framework which parallels (and includes as a limiting case) the differential setting. The integral operators introduced have convolution structures and converge as the horizon of interaction shrinks to zero to the classical gradient, divergence, curl, and Laplacian. Moreover, a Helmholtz-type decomposition holds on the entire \(\mathbb{R}^n\), so general vector fields can be decomposed into (nonlocal) divergence-free and curl-free components. We also identify the kernels of the nonlocal operators and prove additional properties towards building a nonlocal framework suitable for analysis of integro-differential systems.
MSC:
| 47Gxx | Integral, integro-differential, and pseudodifferential operators |
| 26A33 | Fractional derivatives and integrals |
| 35S30 | Fourier integral operators applied to PDEs |
| 41A35 | Approximation by operators (in particular, by integral operators) |
| 45A05 | Linear integral equations |
| 45P05 | Integral operators |
| 46F12 | Integral transforms in distribution spaces |
| 46N20 | Applications of functional analysis to differential and integral equations |
References:
| [1] | Silling, S., Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48, 175-209 (2000) · Zbl 0970.74030 · doi:10.1016/S0022-5096(99)00029-0 |
| [2] | Silling, SA, Linearized theory of peridynamic states, J. Elast., 99, 1, 85-111 (2010) · Zbl 1188.74008 · doi:10.1007/s10659-009-9234-0 |
| [3] | Andreu-Vaillo, F.; Mazón, JM; Rossi, JD; Toledo-Melero, JJ, Nonlocal Diffusion Problems. Volume 165 of Mathematical Surveys and Monographs (2010), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1214.45002 · doi:10.1090/surv/165 |
| [4] | Carrillo, C.; Fife, P., Spatial effects in discrete generation population models, J. Math. Biol., 50, 2, 161-188 (2005) · Zbl 1080.92054 · doi:10.1007/s00285-004-0284-4 |
| [5] | Gilboa, G.; Osher, S., Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7, 3, 1005-1028 (2009) · Zbl 1181.35006 · doi:10.1137/070698592 |
| [6] | Mirman, R., Maxwell’s equations and Helmholtz’s theorem, Am. J. Phys., 33, 6, 503-504 (1965) · doi:10.1119/1.1971745 |
| [7] | Chorin, AJ, A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 135, 2, 118-125 (1997) · Zbl 0899.76283 · doi:10.1006/jcph.1997.5716 |
| [8] | Palit, B., Basu, A., Mandal, M. K.: Applications of the discrete Hodge Helmholtz decomposition to image and video processing. In: International Conference on Pattern Recognition and Machine Intelligence, pp. 497-502. Springer (2005) |
| [9] | Bhatia, H.; Norgard, G.; Pascucci, V.; Bremer, P-T, The Helmholtz-Hodge decomposition-a survey, IEEE Trans. Vis. Comput. Graphics, 19, 8, 1386-1404 (2013) · doi:10.1109/TVCG.2012.316 |
| [10] | Caffarelli, L.; Holden, H.; Karlsen, K., Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, 37-52 (2012), Berlin: Springer, Berlin · Zbl 1266.35060 · doi:10.1007/978-3-642-25361-4_3 |
| [11] | Carpinteri, A.; Cornetti, P.; Sapora, A., A fractional calculus approach to nonlocal elasticity, Eur. Phys. J. Special Topics, 193, 1, 193-204 (2011) · doi:10.1140/epjst/e2011-01391-5 |
| [12] | Du, Q.; Gunzburger, M.; Lehoucq, RB; Zhou, K., A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci., 23, 3, 493-540 (2013) · Zbl 1266.26020 · doi:10.1142/S0218202512500546 |
| [13] | D’Elia, M.; Flores, C.; Li, X.; Radu, P.; Yu, Y., Helmholtz-Hodge decompositions in the nonlocal framework, J. Peridyn. Nonlocal Model., 2, 401-418 (2020) · doi:10.1007/s42102-020-00035-w |
| [14] | Laskin, N.; Tarasov, VE, Nonlocal quantum mechanics: fractional calculus approach, Applications in Physics, 207 (2019), Berlin: De Gruyter, Berlin · doi:10.1515/9783110571721-009 |
| [15] | Lee, H.; Du, Q., Nonlocal gradient operators with a nonspherical interaction neighborhood and their applications, ESAIM: Math. Model. Numer. Anal., 54, 1, 105-128 (2020) · Zbl 1450.76031 · doi:10.1051/m2an/2019053 |
| [16] | Gunzburger, M.; Lehoucq, RB, A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8, 1581-1598 (2010) · Zbl 1210.35057 · doi:10.1137/090766607 |
| [17] | Shankar, R., On a nonlocal extension of differentiation, J. Math. Anal. Appl., 440, 2, 516-528 (2016) · Zbl 1382.45012 · doi:10.1016/j.jmaa.2016.03.054 |
| [18] | Foss, MD; Radu, P.; Voyiadjis, GZ, Bridging local and nonlocal models: Convergence and regularity, Handbook of Nonlocal Continuum Mechanics for Materials and Structures (2019), Cham: Springer International Publishing, Cham |
| [19] | Mengesha, T.; Du, Q., The bond-based peridynamic system with Dirichlet-type volume constraint, Proc. R. Soc. Edinb. Sect. A, 144, 1, 161-186 (2014) · Zbl 1381.35177 · doi:10.1017/S0308210512001436 |
| [20] | Radu, P.; Toundykov, D.; Trageser, J., A nonlocal biharmonic operator and its connection with the classical analogue, Arch. Ration. Mech. Anal., 223, 2, 845-880 (2017) · Zbl 1372.35093 · doi:10.1007/s00205-016-1047-2 |
| [21] | Du, Q.; Gunzburger, M.; Lehoucq, RB; Zhou, K., Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54, 4, 667-696 (2012) · Zbl 1422.76168 · doi:10.1137/110833294 |
| [22] | Radu, P.; Wells, K., A doubly nonlocal Laplace operator and its connection to the classical Laplacian, J. Integr. Equ. Appl., 31, 3, 379-409 (2019) · Zbl 1429.35189 · doi:10.1216/JIE-2019-31-3-379 |
| [23] | Bobaru, F.; Duangpanya, M., The peridynamic formulation for transient heat conduction, Int. J. Heat Mass Transfer, 53, 19-20, 4047-4059 (2010) · Zbl 1194.80010 · doi:10.1016/j.ijheatmasstransfer.2010.05.024 |
| [24] | Foss, M. D.: Nonlocal Poincaré inequalities for integral operators with integrable nonhomogeneous kernels, arXiv preprint arXiv:1911.10292 (2019) |
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