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 & 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. 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. 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, 165 (2010). Google Scholar

[4]

J. Bourgain, H. Brézis and P. Mironescu, Another look at Sobolev spaces,, in, (2001), 439. 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, 57 (2002), 59. 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, 5 (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. 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. 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. 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. 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. 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. Google Scholar

[15]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing,, Multiscale Modeling and Simulation, 7 (2008), 1005. 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. 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, (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. 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. 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. 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. Google Scholar

[23]

S. Silling, O. Weckner, E. Askari and F. Bobaru, Crack nucleation in a peridynamic solid,, Int. J. Fract., 162 (2010), 219. 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. 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. 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. 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, 165 (2010). Google Scholar

[4]

J. Bourgain, H. Brézis and P. Mironescu, Another look at Sobolev spaces,, in, (2001), 439. 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, 57 (2002), 59. 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, 5 (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. 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. 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. 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. 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. 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. Google Scholar

[15]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing,, Multiscale Modeling and Simulation, 7 (2008), 1005. 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. 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, (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. 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. 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. 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. Google Scholar

[23]

S. Silling, O. Weckner, E. Askari and F. Bobaru, Crack nucleation in a peridynamic solid,, Int. J. Fract., 162 (2010), 219. 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. doi: 10.1137/090781267. Google Scholar

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