American Institute of Mathematical Sciences

Advanced
July  2013, 18(5): 1415-1437. doi: 10.3934/dcdsb.2013.18.1415

Analysis of a scalar nonlocal peridynamic model with a sign changing kernel

Tadele Mengesha 1, and  Qiang Du 1,

1. 

Department of Mathematics, Pennsylvania State University, University Park, PA 16802, United States

Received  January 2013 Revised  February 2013 Published  March 2013

In this paper, a scalar peridynamic model is analyzed. The study extends earlier works in the literature on scalar nonlocal diffusion and nonlocal peridynamic models to include a sign changing kernel. We prove the well-posedness of both variational problems with nonlocal constraints and time-dependent equations with or without damping. The analysis is based on some nonlocal Poincaré type inequalities and compactness of the associated nonlocal operators. It also offers careful characterizations of the associated solution spaces such as compact embedding, separability and completeness along with regularity properties of solutions for different types of kernels.
Keywords: well-posedness., nonlocal diffusion, nonlocal operator, fractional Sobolev spaces, nonlocal Poincare inequality, Peridynamic model.
Mathematics Subject Classification: 45A05, 45K05, 47G10, 74G65, 74H20, 74H2.
Citation: Tadele Mengesha, Qiang Du. Analysis of a scalar nonlocal peridynamic model with a sign changing kernel. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1415-1437. doi: 10.3934/dcdsb.2013.18.1415
References:
[1]

B. Aksoylu and T. Mengesha, Results on nonlocal boundary value problems, Numerical Functional Analysis and Optimization, 31 (2010), 1301-1317. doi: 10.1080/01630563.2010.519136.  Google Scholar

[2]

B. Alali and R. Lipton, Multiscale analysis of heterogeneous media in the peridynamic formulation, Journal of Elasticity, 106 (2012), 71-103. doi: 10.1007/s10659-010-9291-4.  Google Scholar

[3]

F. Andreu-Vaillo, J. M. Mazn, Julio D. Rossi and J. J. Toledo-Melero, "Nonlocal Diffusion Problems," American Mathematical Society. Mathematical Surveys and Monographs, 2010. 165.  Google Scholar

[4]

J. Bourgain, H. Brézis and P. Mironescu, Another look at Sobolev spaces, in "Optimal Control and Partial Differential Equations" (Editors, J. L. Menaldi, E. Rofman and A. Sulem), IOS Press (2001), 439-455. A volume in honour of A. Benssoussan's 60th birthday. Google Scholar

[5]

H. Brézis, "Analyse Fonctionnelle. Théorie et Applications," Masson, 1978.  Google Scholar

[6]

H. Brézis, How to recognize constant functions. Connections with Sobolev spaces, Uspekhi Mat. Nauk 57, 59-74 (2002)(Russian). English version: Russian Math Surveys 57 (2002), 693-708, Volume in Honor of M. Vishik. doi: 10.1070/RM2002v057n04ABEH000533.  Google Scholar

[7]

R. Dautray and J-L. Lions, "Mathematical and Numerical Analysis for Science and Technology, Evolution Problems I," 5, Springer-Verlag, 1992. doi: 10.1007/978-3-642-58090-1.  Google Scholar

[8]

K. Dayal and K. Bhattacharya, Kinetics of phase transformations in the peridynamic formulation of continuum mechanics, J. Mech. Phys. Solids, 54 (2006), 1811-1842. doi: 10.1016/j.jmps.2006.04.001.  Google Scholar

[9]

Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, Analysis of the volume-constrained peridynamic Navier equation of linear elasticity, to appear in Journal of Elasticity, 2013. doi: 10.1007/s10659-012-9418-x.  Google Scholar

[10]

Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Mod. Meth. Appl. Sci., 23 (2013), 493-540. doi: 10.1142/S0218202512500546.  Google Scholar

[11]

Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Review, 54 (2012), 667-696. doi: 10.1137/110833294.  Google Scholar

[12]

Q. Du and K. Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory, ESIAM: Math. Mod. Numer. Anal., 45 (2011), 217-234. doi: 10.1051/m2an/2010040.  Google Scholar

[13]

E. Emmrich and O. Weckner, The peridynamic equation and its spatial discretization, Mathematical Modelling and Analysis, 12 (2007), 17-27. doi: 10.3846/1392-6292.2007.12.17-27.  Google Scholar

[14]

E. Emmrich and O. Weckner, On the well-posedness of the linear peridynamic model and its convergence towards the {Navier equation of linear elasticity}, Commun. Math. Sci., 5 (2007), 851-864.  Google Scholar

[15]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling and Simulation, 7 (2008), {1005-1028}. doi: 10.1137/070698592.  Google Scholar

[16]

M. Gunzburger and R. B. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8 (2010), 1581-1598. doi: 10.1137/090766607.  Google Scholar

[17]

T. Mengesha and Q. Du, The bond-based peridynamic system with Dirichlet-type volume constraint, Proceeding of Royal Soc. Edinburgh A, to appear 2013. Google Scholar

[18]

T. Mengesha and Q. Du, Nonlocal constrained value problems for a linear peridynamic Navier equation, preprint, 2013. Google Scholar

[19]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Physics, 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9.  Google Scholar

[20]

S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48 (2000), {175-209}. doi: 10.1016/S0022-5096(99)00029-0.  Google Scholar

[21]

E. D. Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[22]

A. Ponce, An estimate in the spirit of Poincare's inequality, J. Eur. Math. Soc., 6 (2004), {1-15}.  Google Scholar

[23]

S. Silling, O. Weckner, E. Askari and F. Bobaru, Crack nucleation in a peridynamic solid, Int. J. Fract., 162 (2010), 219-227. Google Scholar

[24]

K. Zhou and Q. Du, Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary, SIAM J. Numer. Anal., 48 (2010), 1759-1780. doi: 10.1137/090781267.  Google Scholar

show all references

References:
[1]

B. Aksoylu and T. Mengesha, Results on nonlocal boundary value problems, Numerical Functional Analysis and Optimization, 31 (2010), 1301-1317. doi: 10.1080/01630563.2010.519136.  Google Scholar

[2]

B. Alali and R. Lipton, Multiscale analysis of heterogeneous media in the peridynamic formulation, Journal of Elasticity, 106 (2012), 71-103. doi: 10.1007/s10659-010-9291-4.  Google Scholar

[3]

F. Andreu-Vaillo, J. M. Mazn, Julio D. Rossi and J. J. Toledo-Melero, "Nonlocal Diffusion Problems," American Mathematical Society. Mathematical Surveys and Monographs, 2010. 165.  Google Scholar

[4]

J. Bourgain, H. Brézis and P. Mironescu, Another look at Sobolev spaces, in "Optimal Control and Partial Differential Equations" (Editors, J. L. Menaldi, E. Rofman and A. Sulem), IOS Press (2001), 439-455. A volume in honour of A. Benssoussan's 60th birthday. Google Scholar

[5]

H. Brézis, "Analyse Fonctionnelle. Théorie et Applications," Masson, 1978.  Google Scholar

[6]

H. Brézis, How to recognize constant functions. Connections with Sobolev spaces, Uspekhi Mat. Nauk 57, 59-74 (2002)(Russian). English version: Russian Math Surveys 57 (2002), 693-708, Volume in Honor of M. Vishik. doi: 10.1070/RM2002v057n04ABEH000533.  Google Scholar

[7]

R. Dautray and J-L. Lions, "Mathematical and Numerical Analysis for Science and Technology, Evolution Problems I," 5, Springer-Verlag, 1992. doi: 10.1007/978-3-642-58090-1.  Google Scholar

[8]

K. Dayal and K. Bhattacharya, Kinetics of phase transformations in the peridynamic formulation of continuum mechanics, J. Mech. Phys. Solids, 54 (2006), 1811-1842. doi: 10.1016/j.jmps.2006.04.001.  Google Scholar

[9]

Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, Analysis of the volume-constrained peridynamic Navier equation of linear elasticity, to appear in Journal of Elasticity, 2013. doi: 10.1007/s10659-012-9418-x.  Google Scholar

[10]

Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Mod. Meth. Appl. Sci., 23 (2013), 493-540. doi: 10.1142/S0218202512500546.  Google Scholar

[11]

Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Review, 54 (2012), 667-696. doi: 10.1137/110833294.  Google Scholar

[12]

Q. Du and K. Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory, ESIAM: Math. Mod. Numer. Anal., 45 (2011), 217-234. doi: 10.1051/m2an/2010040.  Google Scholar

[13]

E. Emmrich and O. Weckner, The peridynamic equation and its spatial discretization, Mathematical Modelling and Analysis, 12 (2007), 17-27. doi: 10.3846/1392-6292.2007.12.17-27.  Google Scholar

[14]

E. Emmrich and O. Weckner, On the well-posedness of the linear peridynamic model and its convergence towards the {Navier equation of linear elasticity}, Commun. Math. Sci., 5 (2007), 851-864.  Google Scholar

[15]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling and Simulation, 7 (2008), {1005-1028}. doi: 10.1137/070698592.  Google Scholar

[16]

M. Gunzburger and R. B. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8 (2010), 1581-1598. doi: 10.1137/090766607.  Google Scholar

[17]

T. Mengesha and Q. Du, The bond-based peridynamic system with Dirichlet-type volume constraint, Proceeding of Royal Soc. Edinburgh A, to appear 2013. Google Scholar

[18]

T. Mengesha and Q. Du, Nonlocal constrained value problems for a linear peridynamic Navier equation, preprint, 2013. Google Scholar

[19]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Physics, 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9.  Google Scholar

[20]

S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48 (2000), {175-209}. doi: 10.1016/S0022-5096(99)00029-0.  Google Scholar

[21]

E. D. Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[22]

A. Ponce, An estimate in the spirit of Poincare's inequality, J. Eur. Math. Soc., 6 (2004), {1-15}.  Google Scholar

[23]

S. Silling, O. Weckner, E. Askari and F. Bobaru, Crack nucleation in a peridynamic solid, Int. J. Fract., 162 (2010), 219-227. Google Scholar

[24]

K. Zhou and Q. Du, Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary, SIAM J. Numer. Anal., 48 (2010), 1759-1780. doi: 10.1137/090781267.  Google Scholar

[1]

Boumediene Abdellaoui, Fethi Mahmoudi. An improved Hardy inequality for a nonlocal operator. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1143-1157. doi: 10.3934/dcds.2016.36.1143
[2]

Pablo Raúl Stinga, Chao Zhang. Harnack's inequality for fractional nonlocal equations. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3153-3170. doi: 10.3934/dcds.2013.33.3153
[3]

Dong Li, Xiaoyi Zhang. On a nonlocal aggregation model with nonlinear diffusion. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 301-323. doi: 10.3934/dcds.2010.27.301
[4]

Armel Ovono Andami. From local to nonlocal in a diffusion model. Conference Publications, 2011, 2011 (Special) : 54-60. doi: 10.3934/proc.2011.2011.54
[5]

Elisabeth Logak, Isabelle Passat. An epidemic model with nonlocal diffusion on networks. Networks & Heterogeneous Media, 2016, 11 (4) : 693-719. doi: 10.3934/nhm.2016014
[6]

Meng Zhao, Wantong Li, Yihong Du. The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4599-4620. doi: 10.3934/cpaa.2020208
[7]

Abraham Sylla. Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model. Networks & Heterogeneous Media, 2021, 16 (2) : 221-256. doi: 10.3934/nhm.2021005
[8]

Luc Molinet, Francis Ribaud. On global well-posedness for a class of nonlocal dispersive wave equations. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 657-668. doi: 10.3934/dcds.2006.15.657
[9]

Manas Bhatnagar, Hailiang Liu. Well-posedness and critical thresholds in a nonlocal Euler system with relaxation. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5271-5289. doi: 10.3934/dcds.2021076
[10]

Irene Benedetti, Luisa Malaguti, Valentina Taddei. Nonlocal problems in Hilbert spaces. Conference Publications, 2015, 2015 (special) : 103-111. doi: 10.3934/proc.2015.0103
[11]

Jinrong Wang, Michal Fečkan, Yong Zhou. Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 471-486. doi: 10.3934/eect.2017024
[12]

Anouar Bahrouni, VicenŢiu D. RĂdulescu. On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 379-389. doi: 10.3934/dcdss.2018021
[13]

Keng Deng. On a nonlocal reaction-diffusion population model. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 65-73. doi: 10.3934/dcdsb.2008.9.65
[14]

Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171
[15]

Siwei Duo, Hong Wang, Yanzhi Zhang. A comparative study on nonlocal diffusion operators related to the fractional Laplacian. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 231-256. doi: 10.3934/dcdsb.2018110
[16]

Vincenzo Ambrosio, Giovanni Molica Bisci. Periodic solutions for nonlocal fractional equations. Communications on Pure & Applied Analysis, 2017, 16 (1) : 331-344. doi: 10.3934/cpaa.2017016
[17]

J. García-Melián, Julio D. Rossi. A logistic equation with refuge and nonlocal diffusion. Communications on Pure & Applied Analysis, 2009, 8 (6) : 2037-2053. doi: 10.3934/cpaa.2009.8.2037
[18]

Huimin Liang, Peixuan Weng, Yanling Tian. Bility and traveling wavefronts for a convolution model of mistletoes and birds with nonlocal diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2207-2231. doi: 10.3934/dcdsb.2017093
[19]

Jia-Feng Cao, Wan-Tong Li, Meng Zhao. On a free boundary problem for a nonlocal reaction-diffusion model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4117-4139. doi: 10.3934/dcdsb.2018128
[20]

Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (198)
  • HTML views (0)
  • Cited by (37)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy