-
Previous Article
Some Remarks on regularity criteria of Axially symmetric Navier-Stokes equations
- CPAA Home
- This Issue
-
Next Article
Stability and $ L^{p}$ convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems
Solvability of nonlocal systems related to peridynamics
1. | Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany |
2. | Department of Mathematics, The University of Tennessee Knoxville, 227 Ayres Hall, 1403 Circle Drive, Knoxville, TN 37996, USA |
In this work, we study the Dirichlet problem associated with a strongly coupled system of nonlocal equations. The system of equations comes from a linearization of a model of peridynamics, a nonlocal model of elasticity. It is a nonlocal analogue of the Navier-Lamé system of classical elasticity. The leading operator is an integro-differential operator characterized by a distinctive matrix kernel which is used to couple differences of components of a vector field. The paper's main contributions are proving well-posedness of the system of equations and demonstrating optimal local Sobolev regularity of solutions. We apply Hilbert space techniques for well-posedness. The result holds for systems associated with kernels that give rise to non-symmetric bilinear forms. The regularity result holds for systems with symmetric kernels that may be supported only on a cone. For some specific kernels associated energy spaces are shown to coincide with standard fractional Sobolev spaces.
References:
[1] |
U. Biccari, M. Warma and E. Zuazua,
Addendum: Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17 (2017), 837-839.
doi: 10.1515/ans-2017-6020. |
[2] |
U. Biccari, M. Warma and E. Zuazua,
Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17 (2017), 387-409.
doi: 10.1515/ans-2017-0014. |
[3] |
H. Brezis, Functional analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
[4] |
Z.-Q. Chen,
On notions of harmonicity, Proc. Amer. Math. Soc., 137 (2009), 3497-3510.
doi: 10.1090/S0002-9939-09-09945-6. |
[5] |
M. Cozzi,
Interior regularity of solutions of non-local equations in Sobolev and Nikol'skii spaces, Ann. Mat. Pura Appl., 196 (2017), 555-578.
doi: 10.1007/s10231-016-0586-3. |
[6] |
F. Da Lio,
Fractional harmonic maps into manifolds in odd dimension $ n>1$, Calc. Var. Partial Differential Equations, 48 (2013), 421-445.
doi: 10.1007/s00526-012-0556-6. |
[7] |
F. Da Lio and T. Rivière,
Three-term commutator estimates and the regularity of $ \frac{1}{2}$-harmonic maps into spheres, Anal. PDE, 4 (2011), 149-190.
doi: 10.2140/apde.2011.4.149. |
[8] |
F. Da Lio and A. Schikorra, On regularity theory for n/p-harmonic maps into manifolds, arXiv e-prints, (2017).
doi: 10.1016/j.na.2017.10.001. |
[9] |
H. Dong and D. Kim,
On $ L_p$-estimates for a class of non-local elliptic equations, J. Funct. Anal., 262 (2012), 1166-1199.
doi: 10.1016/j.jfa.2011.11.002. |
[10] |
Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou,
Analysis of the volume-constrained peridynamic Navier equation of linear elasticity, J. Elasticity, 113 (2013), 193-217.
doi: 10.1007/s10659-012-9418-x. |
[11] |
Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou,
A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci., 23 (2013), 493-540.
doi: 10.1142/S0218202512500546. |
[12] |
Q. Du and K. Zhou,
Mathematical analysis for the peridynamic nonlocal continuum theory, ESAIM Math. Model. Numer. Anal., 45 (2011), 217-234.
doi: 10.1051/m2an/2010040. |
[13] |
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.
|
[14] |
M. Felsinger, M. Kassmann and P. Voigt,
The Dirichlet problem for nonlocal operators, Math. Z., 279 (2015), 779-809.
doi: 10.1007/s00209-014-1394-3. |
[15] |
M. Fukushima, Y. Oshima, and M. Takeda,
Dirichlet Forms and Symmetric MArkov Processes, vol. 19 of De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, extended ed., 2011. |
[16] |
M. Fukushima and T. Uemura,
Jump-type Hunt processes generated by lower bounded semi-Dirichlet forms, Ann. Probab., 40 (2012), 858-889.
doi: 10.1214/10-AOP633. |
[17] |
M. Giaquinta and L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs vol. 11 of Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], Edizioni della Normale, Pisa, second ed., 2012.
doi: 10.1007/978-88-7642-443-4. |
[18] |
G. Grubb,
Fractional Laplacians on domains, a development of Hörmander's theory of µ-transmission pseudodifferential operators, Adv. Math., 268 (2015), 478-528.
doi: 10.1016/j.aim.2014.09.018. |
[19] |
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. |
[20] |
Z.-C. Hu, Z.-M. Ma and W. Sun,
On representations of non-symmetric Dirichlet forms, Potential Anal., 32 (2010), 101-131.
doi: 10.1007/s11118-009-9145-5. |
[21] |
M. Kassmann and B. Dyda, Function spaces and extension results for nonlocal Dirichlet problems, arXiv e-prints, (2016). |
[22] |
T. Leonori, I. Peral, A. Primo and F. Soria,
Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.
doi: 10.3934/dcds.2015.35.6031. |
[23] |
Z. Ma, R. Zhu and X. Zhu,
On notions of harmonicity for non-symmetric Dirichlet form, Sci. China Math., 53 (2010), 1407-1420.
doi: 10.1007/s11425-010-4001-z. |
[24] |
T. Mengesha, Fractional Korn and Hardy-type inequalities for vector fields in half space, Communications in Contemporary Mathematics, to appear.
doi: 10.1142/S0219199718500554. |
[25] |
T. Mengesha and Q. Du,
The bond-based peridynamic system with Dirichlet-type volume constraint, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 161-186.
doi: 10.1017/S0308210512001436. |
[26] |
T. Mengesha and Q. Du,
Nonlocal constrained value problems for a linear peridynamic Navier equation, J. Elasticity, 116 (2014), 27-51.
doi: 10.1007/s10659-013-9456-z. |
[27] |
V. Millot and Y. Sire,
On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres, Arch. Ration. Mech. Anal., 215 (2015), 125-210.
doi: 10.1007/s00205-014-0776-3. |
[28] |
X. Ros-Oton,
Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3-26.
|
[29] |
A. Rutkowski,
The Dirichlet problem for nonlocal Lévy-type operators, Publ. Mat., 62 (2018), 213-251.
doi: 10.5565/PUBLMAT6211811. |
[30] |
A. Schikorra,
Regularity of n/2-harmonic maps into spheres, J. Differential Equations, 252 (2012), 1862-1911.
doi: 10.1016/j.jde.2011.08.021. |
[31] |
A. Schikorra,
Lp-gradient harmonic maps into spheres and SO(N), Differential Integral Equations, 28 (2015), 383-408.
|
[32] |
R. L. Schilling and J. Wang, Lower bounded semi-Dirichlet forms associated with Lévy type operators, in Festschrift Masatoshi Fukushima, vol. 17 of Interdiscip. Math. Sci., World Sci. Publ., Hackensack, NJ, 2015,507-526.
doi: 10.1142/9789814596534_0025. |
[33] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
doi: 10.3934/dcds.2013.33.2105. |
[34] |
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. |
[35] |
S. A. Silling,
Linearized theory of peridynamic states, J. Elasticity, 99 (2010), 85-111.
doi: 10.1007/s10659-009-9234-0. |
[36] |
S. A. Silling, M. Epton, O. Weckner, J. Xu, and E. Askari, Peridynamic states and constitutive modeling, J. Elasticity, 88 (2007), 151-184.
doi: 10.1007/s10659-007-9125-1. |
[37] |
E. M. Stein,
Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. |
[38] |
R. Temam and A. Miranville,
Mathematical Modeling in Continuum Mechanics, Cambridge University Press, Cambridge, second ed., 2005.
doi: 10.1017/CBO9780511755422. |
show all references
References:
[1] |
U. Biccari, M. Warma and E. Zuazua,
Addendum: Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17 (2017), 837-839.
doi: 10.1515/ans-2017-6020. |
[2] |
U. Biccari, M. Warma and E. Zuazua,
Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17 (2017), 387-409.
doi: 10.1515/ans-2017-0014. |
[3] |
H. Brezis, Functional analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
[4] |
Z.-Q. Chen,
On notions of harmonicity, Proc. Amer. Math. Soc., 137 (2009), 3497-3510.
doi: 10.1090/S0002-9939-09-09945-6. |
[5] |
M. Cozzi,
Interior regularity of solutions of non-local equations in Sobolev and Nikol'skii spaces, Ann. Mat. Pura Appl., 196 (2017), 555-578.
doi: 10.1007/s10231-016-0586-3. |
[6] |
F. Da Lio,
Fractional harmonic maps into manifolds in odd dimension $ n>1$, Calc. Var. Partial Differential Equations, 48 (2013), 421-445.
doi: 10.1007/s00526-012-0556-6. |
[7] |
F. Da Lio and T. Rivière,
Three-term commutator estimates and the regularity of $ \frac{1}{2}$-harmonic maps into spheres, Anal. PDE, 4 (2011), 149-190.
doi: 10.2140/apde.2011.4.149. |
[8] |
F. Da Lio and A. Schikorra, On regularity theory for n/p-harmonic maps into manifolds, arXiv e-prints, (2017).
doi: 10.1016/j.na.2017.10.001. |
[9] |
H. Dong and D. Kim,
On $ L_p$-estimates for a class of non-local elliptic equations, J. Funct. Anal., 262 (2012), 1166-1199.
doi: 10.1016/j.jfa.2011.11.002. |
[10] |
Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou,
Analysis of the volume-constrained peridynamic Navier equation of linear elasticity, J. Elasticity, 113 (2013), 193-217.
doi: 10.1007/s10659-012-9418-x. |
[11] |
Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou,
A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci., 23 (2013), 493-540.
doi: 10.1142/S0218202512500546. |
[12] |
Q. Du and K. Zhou,
Mathematical analysis for the peridynamic nonlocal continuum theory, ESAIM Math. Model. Numer. Anal., 45 (2011), 217-234.
doi: 10.1051/m2an/2010040. |
[13] |
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.
|
[14] |
M. Felsinger, M. Kassmann and P. Voigt,
The Dirichlet problem for nonlocal operators, Math. Z., 279 (2015), 779-809.
doi: 10.1007/s00209-014-1394-3. |
[15] |
M. Fukushima, Y. Oshima, and M. Takeda,
Dirichlet Forms and Symmetric MArkov Processes, vol. 19 of De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, extended ed., 2011. |
[16] |
M. Fukushima and T. Uemura,
Jump-type Hunt processes generated by lower bounded semi-Dirichlet forms, Ann. Probab., 40 (2012), 858-889.
doi: 10.1214/10-AOP633. |
[17] |
M. Giaquinta and L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs vol. 11 of Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], Edizioni della Normale, Pisa, second ed., 2012.
doi: 10.1007/978-88-7642-443-4. |
[18] |
G. Grubb,
Fractional Laplacians on domains, a development of Hörmander's theory of µ-transmission pseudodifferential operators, Adv. Math., 268 (2015), 478-528.
doi: 10.1016/j.aim.2014.09.018. |
[19] |
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. |
[20] |
Z.-C. Hu, Z.-M. Ma and W. Sun,
On representations of non-symmetric Dirichlet forms, Potential Anal., 32 (2010), 101-131.
doi: 10.1007/s11118-009-9145-5. |
[21] |
M. Kassmann and B. Dyda, Function spaces and extension results for nonlocal Dirichlet problems, arXiv e-prints, (2016). |
[22] |
T. Leonori, I. Peral, A. Primo and F. Soria,
Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.
doi: 10.3934/dcds.2015.35.6031. |
[23] |
Z. Ma, R. Zhu and X. Zhu,
On notions of harmonicity for non-symmetric Dirichlet form, Sci. China Math., 53 (2010), 1407-1420.
doi: 10.1007/s11425-010-4001-z. |
[24] |
T. Mengesha, Fractional Korn and Hardy-type inequalities for vector fields in half space, Communications in Contemporary Mathematics, to appear.
doi: 10.1142/S0219199718500554. |
[25] |
T. Mengesha and Q. Du,
The bond-based peridynamic system with Dirichlet-type volume constraint, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 161-186.
doi: 10.1017/S0308210512001436. |
[26] |
T. Mengesha and Q. Du,
Nonlocal constrained value problems for a linear peridynamic Navier equation, J. Elasticity, 116 (2014), 27-51.
doi: 10.1007/s10659-013-9456-z. |
[27] |
V. Millot and Y. Sire,
On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres, Arch. Ration. Mech. Anal., 215 (2015), 125-210.
doi: 10.1007/s00205-014-0776-3. |
[28] |
X. Ros-Oton,
Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3-26.
|
[29] |
A. Rutkowski,
The Dirichlet problem for nonlocal Lévy-type operators, Publ. Mat., 62 (2018), 213-251.
doi: 10.5565/PUBLMAT6211811. |
[30] |
A. Schikorra,
Regularity of n/2-harmonic maps into spheres, J. Differential Equations, 252 (2012), 1862-1911.
doi: 10.1016/j.jde.2011.08.021. |
[31] |
A. Schikorra,
Lp-gradient harmonic maps into spheres and SO(N), Differential Integral Equations, 28 (2015), 383-408.
|
[32] |
R. L. Schilling and J. Wang, Lower bounded semi-Dirichlet forms associated with Lévy type operators, in Festschrift Masatoshi Fukushima, vol. 17 of Interdiscip. Math. Sci., World Sci. Publ., Hackensack, NJ, 2015,507-526.
doi: 10.1142/9789814596534_0025. |
[33] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
doi: 10.3934/dcds.2013.33.2105. |
[34] |
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. |
[35] |
S. A. Silling,
Linearized theory of peridynamic states, J. Elasticity, 99 (2010), 85-111.
doi: 10.1007/s10659-009-9234-0. |
[36] |
S. A. Silling, M. Epton, O. Weckner, J. Xu, and E. Askari, Peridynamic states and constitutive modeling, J. Elasticity, 88 (2007), 151-184.
doi: 10.1007/s10659-007-9125-1. |
[37] |
E. M. Stein,
Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. |
[38] |
R. Temam and A. Miranville,
Mathematical Modeling in Continuum Mechanics, Cambridge University Press, Cambridge, second ed., 2005.
doi: 10.1017/CBO9780511755422. |
[1] |
James Scott, Tadele Mengesha. A fractional Korn-type inequality. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3315-3343. doi: 10.3934/dcds.2019137 |
[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] |
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 |
[4] |
Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382 |
[5] |
Jian-Wen Peng, Xin-Min Yang. Levitin-Polyak well-posedness of a system of generalized vector variational inequality problems. Journal of Industrial and Management Optimization, 2015, 11 (3) : 701-714. doi: 10.3934/jimo.2015.11.701 |
[6] |
Akram Ben Aissa. Well-posedness and direct internal stability of coupled non-degenrate Kirchhoff system via heat conduction. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 983-993. doi: 10.3934/dcdss.2021106 |
[7] |
Cezar Kondo, Ronaldo Pes. Well-posedness for a coupled system of Kawahara/KdV type equations with polynomials nonlinearities. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2615-2641. doi: 10.3934/cpaa.2022063 |
[8] |
Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563 |
[9] |
Hartmut Pecher. Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1081-1096. doi: 10.3934/cpaa.2012.11.1081 |
[10] |
Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036 |
[11] |
George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417 |
[12] |
Lei Zhang, Bin Liu. Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2655-2685. doi: 10.3934/dcds.2018112 |
[13] |
E. Compaan, N. Tzirakis. Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $ \mathbb{R}^+ $. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3867-3895. doi: 10.3934/dcds.2019156 |
[14] |
Vanessa Barros, Felipe Linares. A remark on the well-posedness of a degenerated Zakharov system. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1259-1274. doi: 10.3934/cpaa.2015.14.1259 |
[15] |
Bilal Al Taki. Global well posedness for the ghost effect system. Communications on Pure and Applied Analysis, 2017, 16 (1) : 345-368. doi: 10.3934/cpaa.2017017 |
[16] |
Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321 |
[17] |
Hyungjin Huh, Bora Moon. Low regularity well-posedness for Gross-Neveu equations. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1903-1913. doi: 10.3934/cpaa.2015.14.1903 |
[18] |
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 |
[19] |
Philippe Ciarlet. Korn's inequalities: The linear vs. the nonlinear case. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 473-483. doi: 10.3934/dcdss.2012.5.473 |
[20] |
Jean-Daniel Djida, Arran Fernandez, Iván Area. Well-posedness results for fractional semi-linear wave equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 569-597. doi: 10.3934/dcdsb.2019255 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]