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

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.
Citation: Tadele Mengesha, Qiang Du. Analysis of a scalar nonlocal peridynamic model with a sign changing kernel. Discrete and 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

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

[23]

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

[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.

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

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

[23]

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

[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.

[1]

Boumediene Abdellaoui, Fethi Mahmoudi. An improved Hardy inequality for a nonlocal operator. Discrete and 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 and 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 and 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 and 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 and 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 and Heterogeneous Media, 2021, 16 (2) : 221-256. doi: 10.3934/nhm.2021005

[8]

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

[9]

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

[10]

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

[11]

Jinrong Wang, Michal Fečkan, Yong Zhou. Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions. Evolution Equations and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (245)
  • HTML views (0)
  • Cited by (37)

Other articles
by authors

[Back to Top]