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August  2021, 41(8): 3985-4012. doi: 10.3934/dcds.2021025

Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics

José A. Carrillo 1, , Bertram Düring 2, , Lisa Maria Kreusser 3, and  Carola-Bibiane Schönlieb 3,

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

Received  December 2019 Revised  December 2020 Published  August 2021 Early access  January 2021

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.

Keywords: Nonlinear diffusion, nonlocal aggregation, pattern formation, stationary states, finite volume methods.
Mathematics Subject Classification: Primary: 35B33, 65M08; Secondary: 35B36, 35Q92, 65M12.
Citation: José A. Carrillo, Bertram Düring, Lisa Maria Kreusser, Carola-Bibiane Schönlieb. Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3985-4012. doi: 10.3934/dcds.2021025
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.

[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. 

[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. 

[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.

[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. 

[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. 

[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.

Figure 1.  Stationary solution to the anisotropic interaction equation (10), obtained with the numerical scheme (33) on a grid of size 50 in each spatial direction and different diffusion coefficients $ \delta $ for the spatially homogeneous tensor field with $ s = (0,1) $ and $ l = (1,0) $ and uniformly distributed initial data on a disc on the computational domain $ [-0.5,0.5]^2 $
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Figure 2.  Stationary solution to the anisotropic interaction equation (10), obtained with the numerical scheme (33) on grids of sizes 50,100 and 200 in each spatial direction for the diffusion coefficient $ \delta = 10^{-10} $ for the spatially homogeneous tensor field with $ s = (0,1) $ and $ l = (1,0) $ and uniformly distributed initial data on a disc on the computational domain $ [-0.5,0.5]^2 $
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Figure 3.  Stationary solution to the anisotropic interaction equation (10), obtained with the numerical scheme (33) on grids of sizes 100 and 200 in each spatial direction for the diffusion coefficient $ \delta = 10^{-10} $ for the spatially homogeneous tensor field with $ s = (0,1) $ and $ l = (1,0) $ and uniformly distributed initial data on the computational domain $ [-0.5,0.5]^2 $
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Figure 4.  Stationary solution to the anisotropic interaction equation (10), obtained with the numerical scheme (33) on a grid of size 200 in each spatial direction and different diffusion coefficients $ \delta $ for the spatially homogeneous tensor field with $ s = (0,1) $ and $ l = (1,0) $ and uniformly distributed initial data on the computational domain $ [-0.5,0.5]^2 $
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Figure 5.  Cross-section of stationary solution to the anisotropic interaction equation (10), obtained with the numerical scheme (33) on a grid of size 200 in each spatial direction and diffusion coefficient $ \delta = 10^{-9} $ for the spatially homogeneous tensor field with $ s = (0,1) $ and $ l = (1,0) $ and uniformly distributed initial data on the computational domain $ [-0.5,0.5]^2 $
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Figure 6.  Stationary solution to the anisotropic interaction equation (10), obtained with the numerical scheme (33) on a grid of size 50 in each spatial direction and diffusion coefficient $ \delta = 10^{-10} $ for different spatially inhomogeneous tensor fields from real fingerprint images and uniformly distributed initial data on the computational domain $ [-0.5,0.5]^2 $
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Figure 7.  Numerical solution to the anisotropic interaction equation (10) after $ n $ iterations for different $ n $, obtained with the numerical scheme (33) on a grid of size 50 in each spatial direction with diffusion coefficient $ \delta = 10^{-10} $ for the spatially inhomogeneous tensor field of part of a fingerprint and uniformly distributed initial data on the computational domain $ [-0.5,0.5]^2 $
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Figure 8.  Stationary solution to the anisotropic interaction equation (10), obtained with the numerical scheme (33) on a grid of size 50 in each spatial direction for different values of the diffusion coefficient $ \delta $ for a given spatially inhomogeneous tensor field and uniformly distributed initial data on the computational domain $ [-0.5,0.5]^2 $
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Figure 9.  Stationary solution to the anisotropic interaction equation (10), obtained with the numerical scheme (33) on a grid of size 50 in each spatial direction, diffusion coefficient $ \delta = 10^{-10} $ and different force rescalings $ \eta $ for a given spatially inhomogeneous tensor field and uniformly distributed initial data on the computational domain $ [-0.5,0.5]^2 $
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