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Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics
1. | Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom |
2. | Mathematics Institute, University of Warwick, Zeeman Building, Coventry CV4 7AL, United Kingdom |
3. | Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom |
In this paper, we study the equilibria of an anisotropic, nonlocal aggregation equation with nonlinear diffusion which does not possess a gradient flow structure. Here, the anisotropy is induced by an underlying tensor field. Anisotropic forces cannot be associated with a potential in general and stationary solutions of anisotropic aggregation equations generally cannot be regarded as minimizers of an energy functional. We derive equilibrium conditions for stationary line patterns in the setting of spatially homogeneous tensor fields. The stationary solutions can be regarded as the minimizers of a regularised energy functional depending on a scalar potential. A dimension reduction from the two- to the one-dimensional setting allows us to study the associated one-dimensional problem instead of the two-dimensional setting. We establish $ \Gamma $-convergence of the regularised energy functionals as the diffusion coefficient vanishes, and prove the convergence of minimisers of the regularised energy functional to minimisers of the non-regularised energy functional. Further, we investigate properties of stationary solutions on the torus, based on known results in one spatial dimension. Finally, we prove weak convergence of a numerical scheme for the numerical solution of the anisotropic, nonlocal aggregation equation with nonlinear diffusion and any underlying tensor field, and show numerical results.
References:
[1] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics, Birkhäuser, 2005. |
[2] |
D. Balagué, J. A. Carrillo, T. Laurent and G. Raoul,
Dimensionality of local minimizers of the interaction energy, Arch. Ration. Mech. Anal., 209 (2013), 1055-1088.
doi: 10.1007/s00205-013-0644-6. |
[3] |
D. Balagué, J. A. Carrillo, T. Laurent and G. Raoul,
Nonlocal interactions by repulsive-attractive potentials: radial ins/stability, Phys. D, 260 (2013), 5-25.
doi: 10.1016/j.physd.2012.10.002. |
[4] |
J. Bedrossian,
Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion, Appl. Math. Lett., 24 (2011), 1927-1932.
doi: 10.1016/j.aml.2011.05.022. |
[5] |
A. L. Bertozzi, H. Sun, T. Kolokolnikov, D. Uminsky and J. H. von Brecht,
Ring patterns and their bifurcations in a nonlocal model of biological swarms, Commun. Math. Sci., 13 (2015), 955-985.
doi: 10.4310/CMS.2015.v13.n4.a6. |
[6] |
P. Billingsley, Weak Convergence of Measures: Applications in Probability, Society for Industrial and Applied Mathematics, Philadelphia, 1971. |
[7] |
M. Burger, M. Di Francesco and M. Franek,
Stationary states of quadratic diffusion equations with long-range attraction, Commun. Math. Sci., 11 (2013), 709-738.
doi: 10.4310/CMS.2013.v11.n3.a3. |
[8] |
M. Burger and M. DiFrancesco,
Large time behavior of nonlocal aggregation models with nonlinear diffusion, Netw. Heterog. Media, 3 (2008), 749-785.
doi: 10.3934/nhm.2008.3.749. |
[9] |
M. Burger, B. Düring, L. M. Kreusser, P. A. Markowich and C.-B. Schönlieb,
Pattern formation of a nonlocal, anisotropic interaction model, Math. Models Methods Appl. Sci., 28 (2018), 409-451.
doi: 10.1142/S0218202518500112. |
[10] |
M. Burger, R. Fetecau and Y. Huang,
Stationary states and asymptotic behavior of aggregation models with nonlinear local repulsion, SIAM J. Appl. Dyn. Syst., 13 (2014), 397-424.
doi: 10.1137/130923786. |
[11] |
J. A. Cañizo, J. A. Carrillo and F. S. Patacchini,
Existence of compactly supported global minimisers for the interaction energy, Arch. Ration. Mech. Anal., 217 (2015), 1197-1217.
doi: 10.1007/s00205-015-0852-3. |
[12] |
J. A. Carrillo, K. Craig and Y. Yao, Aggregation-diffusion equations: dynamics, asymptotics, and singular limits, in Active Particles, Volume 2: Advances in Theory, Models, and Applications (eds. N. Bellomo, P. Degond and E. Tadmor), Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, (2019), 65–108. |
[13] |
J. A. Carrillo, A. Chertock and Y. Huang,
A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys., 17 (2015), 233-258.
doi: 10.4208/cicp.160214.010814a. |
[14] |
J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and wasserstein distances, in Collective Dynamics from Bacteria to Crowds: An Excursion Through Modeling, Analysis and Simulation, Springer Vienna, Vienna, 553 (2014), 1–46.
doi: 10.1007/978-3-7091-1785-9_1. |
[15] |
J. A. Carrillo, M. G. Delgadino and F. S. Patacchini,
Existence of ground states for aggregation-diffusion equations, Analysis and Applications, 17 (2019), 393-423.
doi: 10.1142/S0219530518500276. |
[16] |
J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepčev,
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271.
doi: 10.1215/00127094-2010-211. |
[17] |
J. A. Carrillo, B. Düring, L. M. Kreusser and C.-B. Schönlieb,
Stability analysis of line patterns of an anisotropic interaction model, SIAM J. Appl. Dyn. Syst., 18 (2019), 1798-1845.
doi: 10.1137/18M1181638. |
[18] |
J. A. Carrillo, F. James, F. Lagoutière and N. Vauchelet,
The filippov characteristic flow for the aggregation equation with mildly singular potentials, J. Differential Equations, 260 (2016), 304-338.
doi: 10.1016/j.jde.2015.08.048. |
[19] |
M. G. Delgadino, X. Yan and Y. Yao, Uniqueness and non-uniqueness of steady states of aggregation-diffusion equations, Comm. Pure Appl. Math., to Appear arXiv: 1908.09782. Google Scholar |
[20] |
B. Düring, C. Gottschlich, S. Huckemann, L. M. Kreusser and C.-B. Schönlieb,
An anisotropic interaction model for simulating fingerprints, J. Math. Biol., 78 (2019), 2171-2206.
doi: 10.1007/s00285-019-01338-3. |
[21] |
F. Golse, On the dynamics of large particle systems in the mean field limit, in Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Springer International Publishing, Cham, 3 (2016), 1–144.
doi: 10.1007/978-3-319-26883-5_1. |
[22] |
C. Gottschlich, P. Mihǎilescu and A. Munk,
Robust orientation field estimation and extrapolation using semilocal line sensors, IEEE Transactions on Information Forensics and Security, 4 (2009), 802-811.
doi: 10.1109/TIFS.2009.2033219. |
[23] |
F. James and N. Vauchelet,
Chemotaxis: From kinetic equations to aggregate dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101-127.
doi: 10.1007/s00030-012-0155-4. |
[24] |
F. James and N. Vauchelet,
Numerical methods for one-dimensional aggregation equations, SIAM J. Numer. Anal., 53 (2015), 895-916.
doi: 10.1137/140959997. |
[25] |
D.-K. Kim and K. A. Holbrook,
The appearance, density, and distribution of merkel cells in human embryonic and fetal skin: Their relation to sweat gland and hair follicle development, Journal of Investigative Dermatology, 104 (1995), 411-416.
doi: 10.1111/1523-1747.ep12665903. |
[26] |
M. Kücken and A. Newell, A model for fingerprint formation, Europhysics Letters, 68 (2004), 141-146. Google Scholar |
[27] |
M. Kücken and A. Newell,
Fingerprint formation, Journal of Theoretical Biology, 235 (2005), 71-83.
doi: 10.1016/j.jtbi.2004.12.020. |
[28] |
M. Kücken and C. Champod, Merkel cells and the individuality of friction ridge skin, J. Theoret. Biol., 317 (2013), 229-237. Google Scholar |
[29] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations. the locally compact case, part 1, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[30] |
A. Mogilner and L. Edelstein-Keshet,
A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.
doi: 10.1007/s002850050158. |
[31] |
R. Simione, D. Slepčev and I. Topaloglu,
Existence of ground states of nonlocal-interaction energies, J. Stat. Phys., 159 (2015), 972-986.
doi: 10.1007/s10955-015-1215-z. |
[32] |
M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, vol. 34 of A Series of Modern Surveys in Mathematics, Springer-Verlag Berlin Heidelberg, 2000.
doi: 10.1007/978-3-662-04194-9. |
[33] |
C. M. Topaz, A. L. Bertozzi and M. A. Lewis,
A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.
doi: 10.1007/s11538-006-9088-6. |
[34] |
A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Process: With Applications to Statistics, Springer Series in Statistics, Springer, 1996.
doi: 10.1007/978-1-4757-2545-2. |
show all references
References:
[1] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics, Birkhäuser, 2005. |
[2] |
D. Balagué, J. A. Carrillo, T. Laurent and G. Raoul,
Dimensionality of local minimizers of the interaction energy, Arch. Ration. Mech. Anal., 209 (2013), 1055-1088.
doi: 10.1007/s00205-013-0644-6. |
[3] |
D. Balagué, J. A. Carrillo, T. Laurent and G. Raoul,
Nonlocal interactions by repulsive-attractive potentials: radial ins/stability, Phys. D, 260 (2013), 5-25.
doi: 10.1016/j.physd.2012.10.002. |
[4] |
J. Bedrossian,
Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion, Appl. Math. Lett., 24 (2011), 1927-1932.
doi: 10.1016/j.aml.2011.05.022. |
[5] |
A. L. Bertozzi, H. Sun, T. Kolokolnikov, D. Uminsky and J. H. von Brecht,
Ring patterns and their bifurcations in a nonlocal model of biological swarms, Commun. Math. Sci., 13 (2015), 955-985.
doi: 10.4310/CMS.2015.v13.n4.a6. |
[6] |
P. Billingsley, Weak Convergence of Measures: Applications in Probability, Society for Industrial and Applied Mathematics, Philadelphia, 1971. |
[7] |
M. Burger, M. Di Francesco and M. Franek,
Stationary states of quadratic diffusion equations with long-range attraction, Commun. Math. Sci., 11 (2013), 709-738.
doi: 10.4310/CMS.2013.v11.n3.a3. |
[8] |
M. Burger and M. DiFrancesco,
Large time behavior of nonlocal aggregation models with nonlinear diffusion, Netw. Heterog. Media, 3 (2008), 749-785.
doi: 10.3934/nhm.2008.3.749. |
[9] |
M. Burger, B. Düring, L. M. Kreusser, P. A. Markowich and C.-B. Schönlieb,
Pattern formation of a nonlocal, anisotropic interaction model, Math. Models Methods Appl. Sci., 28 (2018), 409-451.
doi: 10.1142/S0218202518500112. |
[10] |
M. Burger, R. Fetecau and Y. Huang,
Stationary states and asymptotic behavior of aggregation models with nonlinear local repulsion, SIAM J. Appl. Dyn. Syst., 13 (2014), 397-424.
doi: 10.1137/130923786. |
[11] |
J. A. Cañizo, J. A. Carrillo and F. S. Patacchini,
Existence of compactly supported global minimisers for the interaction energy, Arch. Ration. Mech. Anal., 217 (2015), 1197-1217.
doi: 10.1007/s00205-015-0852-3. |
[12] |
J. A. Carrillo, K. Craig and Y. Yao, Aggregation-diffusion equations: dynamics, asymptotics, and singular limits, in Active Particles, Volume 2: Advances in Theory, Models, and Applications (eds. N. Bellomo, P. Degond and E. Tadmor), Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, (2019), 65–108. |
[13] |
J. A. Carrillo, A. Chertock and Y. Huang,
A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys., 17 (2015), 233-258.
doi: 10.4208/cicp.160214.010814a. |
[14] |
J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and wasserstein distances, in Collective Dynamics from Bacteria to Crowds: An Excursion Through Modeling, Analysis and Simulation, Springer Vienna, Vienna, 553 (2014), 1–46.
doi: 10.1007/978-3-7091-1785-9_1. |
[15] |
J. A. Carrillo, M. G. Delgadino and F. S. Patacchini,
Existence of ground states for aggregation-diffusion equations, Analysis and Applications, 17 (2019), 393-423.
doi: 10.1142/S0219530518500276. |
[16] |
J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepčev,
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271.
doi: 10.1215/00127094-2010-211. |
[17] |
J. A. Carrillo, B. Düring, L. M. Kreusser and C.-B. Schönlieb,
Stability analysis of line patterns of an anisotropic interaction model, SIAM J. Appl. Dyn. Syst., 18 (2019), 1798-1845.
doi: 10.1137/18M1181638. |
[18] |
J. A. Carrillo, F. James, F. Lagoutière and N. Vauchelet,
The filippov characteristic flow for the aggregation equation with mildly singular potentials, J. Differential Equations, 260 (2016), 304-338.
doi: 10.1016/j.jde.2015.08.048. |
[19] |
M. G. Delgadino, X. Yan and Y. Yao, Uniqueness and non-uniqueness of steady states of aggregation-diffusion equations, Comm. Pure Appl. Math., to Appear arXiv: 1908.09782. Google Scholar |
[20] |
B. Düring, C. Gottschlich, S. Huckemann, L. M. Kreusser and C.-B. Schönlieb,
An anisotropic interaction model for simulating fingerprints, J. Math. Biol., 78 (2019), 2171-2206.
doi: 10.1007/s00285-019-01338-3. |
[21] |
F. Golse, On the dynamics of large particle systems in the mean field limit, in Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Springer International Publishing, Cham, 3 (2016), 1–144.
doi: 10.1007/978-3-319-26883-5_1. |
[22] |
C. Gottschlich, P. Mihǎilescu and A. Munk,
Robust orientation field estimation and extrapolation using semilocal line sensors, IEEE Transactions on Information Forensics and Security, 4 (2009), 802-811.
doi: 10.1109/TIFS.2009.2033219. |
[23] |
F. James and N. Vauchelet,
Chemotaxis: From kinetic equations to aggregate dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101-127.
doi: 10.1007/s00030-012-0155-4. |
[24] |
F. James and N. Vauchelet,
Numerical methods for one-dimensional aggregation equations, SIAM J. Numer. Anal., 53 (2015), 895-916.
doi: 10.1137/140959997. |
[25] |
D.-K. Kim and K. A. Holbrook,
The appearance, density, and distribution of merkel cells in human embryonic and fetal skin: Their relation to sweat gland and hair follicle development, Journal of Investigative Dermatology, 104 (1995), 411-416.
doi: 10.1111/1523-1747.ep12665903. |
[26] |
M. Kücken and A. Newell, A model for fingerprint formation, Europhysics Letters, 68 (2004), 141-146. Google Scholar |
[27] |
M. Kücken and A. Newell,
Fingerprint formation, Journal of Theoretical Biology, 235 (2005), 71-83.
doi: 10.1016/j.jtbi.2004.12.020. |
[28] |
M. Kücken and C. Champod, Merkel cells and the individuality of friction ridge skin, J. Theoret. Biol., 317 (2013), 229-237. Google Scholar |
[29] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations. the locally compact case, part 1, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[30] |
A. Mogilner and L. Edelstein-Keshet,
A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.
doi: 10.1007/s002850050158. |
[31] |
R. Simione, D. Slepčev and I. Topaloglu,
Existence of ground states of nonlocal-interaction energies, J. Stat. Phys., 159 (2015), 972-986.
doi: 10.1007/s10955-015-1215-z. |
[32] |
M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, vol. 34 of A Series of Modern Surveys in Mathematics, Springer-Verlag Berlin Heidelberg, 2000.
doi: 10.1007/978-3-662-04194-9. |
[33] |
C. M. Topaz, A. L. Bertozzi and M. A. Lewis,
A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.
doi: 10.1007/s11538-006-9088-6. |
[34] |
A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Process: With Applications to Statistics, Springer Series in Statistics, Springer, 1996.
doi: 10.1007/978-1-4757-2545-2. |









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