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April  2019, 24(4): 1815-1842. doi: 10.3934/dcdsb.2018238

Swarming in domains with boundaries: Approximation and regularization by nonlinear diffusion

Razvan C. Fetecau 1,, , Mitchell Kovacic 1, and  Ihsan Topaloglu 2,

1. 

Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC V5A 1S6, Canada
2. 

Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, 1015 Floyd Ave, Richmond, VA 23219, USA

* Corresponding author: R. C. Fetecau

Received  November 2017 Revised  April 2018 Published  April 2019 Early access  August 2018

Fund Project: The first author is supported by NSERC grant RGPIN-341834.

We consider an aggregation model with nonlinear diffusion in domains with boundaries and investigate the zero diffusion limit of its solutions. We establish the convergence of weak solutions for fixed times, as well as the convergence of energy minimizers in this limit. Numerical simulations that support the analytical results are presented. A second key scope of the numerical studies is to demonstrate that adding small nonlinear diffusion rectifies a flaw of the plain aggregation model in domains with boundaries, which is to evolve into unstable equilibria (non-minimizers of the energy).

Keywords: Swarm equilibria, energy minimizers, gradient flow, attractors, nonlinear diffusion, nonsmooth dynamics.
Mathematics Subject Classification: Primary: 35A15, 35B35; Secondary: 35B40, 35Q92, 45K05.
Citation: Razvan C. Fetecau, Mitchell Kovacic, Ihsan Topaloglu. Swarming in domains with boundaries: Approximation and regularization by nonlinear diffusion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1815-1842. doi: 10.3934/dcdsb.2018238
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005.

[2]

J. Bedrossian, N. Rodrìguez and A. L. Bertozzi, Local and global well-posedness for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion, Nonlinearity, 24 (2011), 1683-1714.  doi: 10.1088/0951-7715/24/6/001.

[3]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393.  doi: 10.1007/s002110050002.

[4]

A. J. Bernoff and C. M. Topaz, A primer of swarm equilibria, SIAM J. Appl. Dyn. Syst., 10 (2011), 212-250.  doi: 10.1137/100804504.

[5]

A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710.  doi: 10.1088/0951-7715/22/3/009.

[6]

A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in $\bf{R}^n$, Comm. Math. Phys., 274 (2007), 717-735.  doi: 10.1007/s00220-007-0288-1.

[7]

A. L. Bertozzi and D. Slepčev, Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion, Commun. Pure Appl. Anal., 9 (2010), 1617-1637.  doi: 10.3934/cpaa.2010.9.1617.

[8]

P. Billingsley, Weak Convergence of Measures: Applications in Probability, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1971.

[9]

M. Bodnar and J. J. L. Velazquez, An integro-differential equation arising as a limit of individual cell-based models, J. Differential Equations, 222 (2006), 341-380.  doi: 10.1016/j.jde.2005.07.025.

[10]

M. Burger, R. C. 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]

M. Burger and M. D. Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Netw. Heterog. Media, 3 (2008), 749-785.  doi: 10.3934/nhm.2008.3.749.

[12]

M. Burger, M. Di Francesco and M. Franek, Stationary states of quadratic diffusion equations with long-range attraction, Comm. Math. Sci., 11 (2013), 709-738.  doi: 10.4310/CMS.2013.v11.n3.a3.

[13]

J. A. Carrillo, K. Craig and F. S. Patacchini, A blob method for diffusion, arXiv preprint, arXiv: 1709.09195, 2017.

[14]

J. A. Carrillo, S. Hittmeir, B. Volzone and Y. Yao, Nonlinear aggregation-diffusion equations: Radial symmetry and long time asymptotics, arXiv preprint, arXiv: 1603.07767, 2016.

[15]

J. A. Carrillo, D. Slepčev and L. Wu, Nonlocal-interaction equations on uniformly prox-regular sets, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 1209-1247. 

[16]

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.

[17]

J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263.  doi: 10.1007/s00205-005-0386-1.

[18]

L. Chayes, I. Kim and Y. Yao, An aggregation equation with degenerate diffusion: Qualitative property of solutions, SIAM J. Math. Anal., 45 (2013), 2995-3018.  doi: 10.1137/120874965.

[19]

R. Choksi, R. C. Fetecau and I. Topaloglu, On minimizers of interaction functionals with competing attractive and repulsive potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1283-1305.  doi: 10.1016/j.anihpc.2014.09.004.

[20]

K. Craig, The exponential formula for the Wasserstein metric, ESAIM Control Optim. Calc. Var., 22 (2016), 169-187.  doi: 10.1051/cocv/2014069.

[21]

K. Craig, Nonconvex gradient flow in the Wasserstein metric and applications to constrained nonlocal interactions, Proc. Lond. Math. Soc. (3), 114 (2017), 60-102.  doi: 10.1112/plms.12005.

[22]

K. Craig and I. Topaloglu, Convergence of regularized nonlocal interaction energies, SIAM J. Math. Anal., 48 (2016), 34-60.  doi: 10.1137/15M1013882.

[23]

Q. Du and P. Zhang, Existence of weak solutions to some vortex density models, SIAM J. Math. Anal., 34 (2003), 1279-1299 (electronic).  doi: 10.1137/S0036141002408009.

[24]

J. H. M. Evers and T. Kolokolnikov, Metastable states for an aggregation model with noise, SIAM J. Appl. Dyn. Syst., 15 (2016), 2213-2226.  doi: 10.1137/16M1069006.

[25]

K. Fellner and G. Raoul, Stable stationary states of non-local interaction equations, Math. Models Methods Appl. Sci., 20 (2010), 2267-2291.  doi: 10.1142/S0218202510004921.

[26]

R. C. Fetecau and Y. Huang, Equilibria of biological aggregations with nonlocal repulsive-attractive interactions, Phys. D, 260 (2013), 49-64.  doi: 10.1016/j.physd.2012.11.004.

[27]

R. C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.  doi: 10.1088/0951-7715/24/10/002.

[28]

R. C. Fetecau and M. Kovacic, Swarm equilibria in domains with boundaries, SIAM Journal on Applied Dynamical Systems, 16 (2017), 1260-1308.  doi: 10.1137/17M1123900.

[29]

D. D. Holm and V. Putkaradze, Aggregation of finite-size particles with variable mobility, Phys Rev Lett., 95 (2005), 226106. doi: 10.1103/PhysRevLett.95.226106.

[30]

Y. Huang and A. L. Bertozzi, Self-similar blowup solutions to an aggregation equation in $\mathbb{R}^n$, Phys. D, 260 (2013), 26-48.  doi: 10.1137/090774495.

[31]

B. D. Hughes and K. Fellner, Continuum models of cohesive stochastic swarms: the effect of motility on aggregation patterns, Phys. D, 260 (2013), 26-48.  doi: 10.1016/j.physd.2013.05.001.

[32]

M. Ji and M. Egerstedt, Distributed coordination control of multi-agent systems while preserving connectedness, IEEE Trans. Robot., 23 (2007), 693-703. 

[33]

G. Kaib, Stationary states of an aggregation equation with degenerate diffusion and bounded attractive potential, SIAM J. Math. Anal., 49 (2017), 272-296.  doi: 10.1137/16M1072450.

[34]

T. Kolokolnikov, H. Sun, D. Uminsky and A. L. Bertozzi, A theory of complex patterns arising from 2D particle interactions, Phys. Rev. E, Rapid Communications, 84 (2011), 015203(R).

[35]

M. Kovacic, Swarm Dynamics and Equilibria in Domains with Boundaries, PhD thesis, Simon Fraser University, 2018.

[36]

A. J. Leverentz, C. M. Topaz and A. J. Bernoff, Asymptotic dynamics of attractive-repulsive swarms, SIAM J. Appl. Dyn. Syst., 8 (2009), 880-908.  doi: 10.1137/090749037.

[37]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.

[38]

R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., 80 (1995), 309-323.  doi: 10.1215/S0012-7094-95-08013-2.

[39]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158.

[40]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.

[41]

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.

[42]

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, third edition, 2000. doi: 10.1007/978-3-662-04194-9.

[43]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Bio., 68, (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.

[44]

C. M. Topaz, M. R. D'Orsogna, L. Edelstein-Keshet and A. J. Bernoff, Locust dynamics: Behavioral phase change and swarming, PLoS Comput. Biol., 8 (2012), e1002642, 11pp. doi: 10.1371/journal.pcbi.1002642.

[45]

G. Toscani, One-dimensional kinetic models of granular flows, M2AN Math. Model. Numer. Anal., 34 (2000), 1277-1291.  doi: 10.1051/m2an:2000127.

[46]

A. W. van der Vaart and J. Wellner, Weak Convergence and Empirical Processes: With Applications to Statistics, Springer Series in Statistics. Springer, 1996. doi: 10.1007/978-1-4757-2545-2.

[47]

C. Villani, Topics in Optimal Transportation volume 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

[48]

L. Wu and D. Slepčev, Nonlocal interaction equations in environments with heterogeneities and boundaries, Comm. Partial Differential Equations, 40 (2015), 1241-1281.  doi: 10.1080/03605302.2015.1015033.

[49]

Y. Zhang, On continuity equations in space-time domains, arXiv preprint, arXiv: 1701.06237, 2017

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 ETH Zürich. Birkhäuser Verlag, Basel, 2005.

[2]

J. Bedrossian, N. Rodrìguez and A. L. Bertozzi, Local and global well-posedness for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion, Nonlinearity, 24 (2011), 1683-1714.  doi: 10.1088/0951-7715/24/6/001.

[3]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393.  doi: 10.1007/s002110050002.

[4]

A. J. Bernoff and C. M. Topaz, A primer of swarm equilibria, SIAM J. Appl. Dyn. Syst., 10 (2011), 212-250.  doi: 10.1137/100804504.

[5]

A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710.  doi: 10.1088/0951-7715/22/3/009.

[6]

A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in $\bf{R}^n$, Comm. Math. Phys., 274 (2007), 717-735.  doi: 10.1007/s00220-007-0288-1.

[7]

A. L. Bertozzi and D. Slepčev, Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion, Commun. Pure Appl. Anal., 9 (2010), 1617-1637.  doi: 10.3934/cpaa.2010.9.1617.

[8]

P. Billingsley, Weak Convergence of Measures: Applications in Probability, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1971.

[9]

M. Bodnar and J. J. L. Velazquez, An integro-differential equation arising as a limit of individual cell-based models, J. Differential Equations, 222 (2006), 341-380.  doi: 10.1016/j.jde.2005.07.025.

[10]

M. Burger, R. C. 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]

M. Burger and M. D. Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Netw. Heterog. Media, 3 (2008), 749-785.  doi: 10.3934/nhm.2008.3.749.

[12]

M. Burger, M. Di Francesco and M. Franek, Stationary states of quadratic diffusion equations with long-range attraction, Comm. Math. Sci., 11 (2013), 709-738.  doi: 10.4310/CMS.2013.v11.n3.a3.

[13]

J. A. Carrillo, K. Craig and F. S. Patacchini, A blob method for diffusion, arXiv preprint, arXiv: 1709.09195, 2017.

[14]

J. A. Carrillo, S. Hittmeir, B. Volzone and Y. Yao, Nonlinear aggregation-diffusion equations: Radial symmetry and long time asymptotics, arXiv preprint, arXiv: 1603.07767, 2016.

[15]

J. A. Carrillo, D. Slepčev and L. Wu, Nonlocal-interaction equations on uniformly prox-regular sets, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 1209-1247. 

[16]

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.

[17]

J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263.  doi: 10.1007/s00205-005-0386-1.

[18]

L. Chayes, I. Kim and Y. Yao, An aggregation equation with degenerate diffusion: Qualitative property of solutions, SIAM J. Math. Anal., 45 (2013), 2995-3018.  doi: 10.1137/120874965.

[19]

R. Choksi, R. C. Fetecau and I. Topaloglu, On minimizers of interaction functionals with competing attractive and repulsive potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1283-1305.  doi: 10.1016/j.anihpc.2014.09.004.

[20]

K. Craig, The exponential formula for the Wasserstein metric, ESAIM Control Optim. Calc. Var., 22 (2016), 169-187.  doi: 10.1051/cocv/2014069.

[21]

K. Craig, Nonconvex gradient flow in the Wasserstein metric and applications to constrained nonlocal interactions, Proc. Lond. Math. Soc. (3), 114 (2017), 60-102.  doi: 10.1112/plms.12005.

[22]

K. Craig and I. Topaloglu, Convergence of regularized nonlocal interaction energies, SIAM J. Math. Anal., 48 (2016), 34-60.  doi: 10.1137/15M1013882.

[23]

Q. Du and P. Zhang, Existence of weak solutions to some vortex density models, SIAM J. Math. Anal., 34 (2003), 1279-1299 (electronic).  doi: 10.1137/S0036141002408009.

[24]

J. H. M. Evers and T. Kolokolnikov, Metastable states for an aggregation model with noise, SIAM J. Appl. Dyn. Syst., 15 (2016), 2213-2226.  doi: 10.1137/16M1069006.

[25]

K. Fellner and G. Raoul, Stable stationary states of non-local interaction equations, Math. Models Methods Appl. Sci., 20 (2010), 2267-2291.  doi: 10.1142/S0218202510004921.

[26]

R. C. Fetecau and Y. Huang, Equilibria of biological aggregations with nonlocal repulsive-attractive interactions, Phys. D, 260 (2013), 49-64.  doi: 10.1016/j.physd.2012.11.004.

[27]

R. C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.  doi: 10.1088/0951-7715/24/10/002.

[28]

R. C. Fetecau and M. Kovacic, Swarm equilibria in domains with boundaries, SIAM Journal on Applied Dynamical Systems, 16 (2017), 1260-1308.  doi: 10.1137/17M1123900.

[29]

D. D. Holm and V. Putkaradze, Aggregation of finite-size particles with variable mobility, Phys Rev Lett., 95 (2005), 226106. doi: 10.1103/PhysRevLett.95.226106.

[30]

Y. Huang and A. L. Bertozzi, Self-similar blowup solutions to an aggregation equation in $\mathbb{R}^n$, Phys. D, 260 (2013), 26-48.  doi: 10.1137/090774495.

[31]

B. D. Hughes and K. Fellner, Continuum models of cohesive stochastic swarms: the effect of motility on aggregation patterns, Phys. D, 260 (2013), 26-48.  doi: 10.1016/j.physd.2013.05.001.

[32]

M. Ji and M. Egerstedt, Distributed coordination control of multi-agent systems while preserving connectedness, IEEE Trans. Robot., 23 (2007), 693-703. 

[33]

G. Kaib, Stationary states of an aggregation equation with degenerate diffusion and bounded attractive potential, SIAM J. Math. Anal., 49 (2017), 272-296.  doi: 10.1137/16M1072450.

[34]

T. Kolokolnikov, H. Sun, D. Uminsky and A. L. Bertozzi, A theory of complex patterns arising from 2D particle interactions, Phys. Rev. E, Rapid Communications, 84 (2011), 015203(R).

[35]

M. Kovacic, Swarm Dynamics and Equilibria in Domains with Boundaries, PhD thesis, Simon Fraser University, 2018.

[36]

A. J. Leverentz, C. M. Topaz and A. J. Bernoff, Asymptotic dynamics of attractive-repulsive swarms, SIAM J. Appl. Dyn. Syst., 8 (2009), 880-908.  doi: 10.1137/090749037.

[37]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.

[38]

R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., 80 (1995), 309-323.  doi: 10.1215/S0012-7094-95-08013-2.

[39]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158.

[40]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.

[41]

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.

[42]

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, third edition, 2000. doi: 10.1007/978-3-662-04194-9.

[43]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Bio., 68, (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.

[44]

C. M. Topaz, M. R. D'Orsogna, L. Edelstein-Keshet and A. J. Bernoff, Locust dynamics: Behavioral phase change and swarming, PLoS Comput. Biol., 8 (2012), e1002642, 11pp. doi: 10.1371/journal.pcbi.1002642.

[45]

G. Toscani, One-dimensional kinetic models of granular flows, M2AN Math. Model. Numer. Anal., 34 (2000), 1277-1291.  doi: 10.1051/m2an:2000127.

[46]

A. W. van der Vaart and J. Wellner, Weak Convergence and Empirical Processes: With Applications to Statistics, Springer Series in Statistics. Springer, 1996. doi: 10.1007/978-1-4757-2545-2.

[47]

C. Villani, Topics in Optimal Transportation volume 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

[48]

L. Wu and D. Slepčev, Nonlocal interaction equations in environments with heterogeneities and boundaries, Comm. Partial Differential Equations, 40 (2015), 1241-1281.  doi: 10.1080/03605302.2015.1015033.

[49]

Y. Zhang, On continuity equations in space-time domains, arXiv preprint, arXiv: 1701.06237, 2017

Figure 1.  Simulations with potential (22) showing early time dynamics. (a) Snapshots of the diffusive model (1) with $\nu = 10^{-7}$. An insert has been included to show the layer of mass near the origin. (b) Snapshots of the plain aggregation model (2). Concentrations at the origin are represented as circle, square, and diamond markers for $t = 0.5$, $t = 1$, and $t = 5$ respectively. The masses of concentrations have been magnified 10 times for clarity
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Figure 2.  Simulations with potential (22) showing the first mass transfer from the boundary to the free swarm in the diffusive model. (a) Snapshots of the diffusive model (1) with $\nu = 10^{-7}$. (b) Snapshots of the plain aggregation model (2). Concentrations are represented as circle and square markers for $t = 10.9$ and $t = 12.5$ respectively. The masses of concentrations have been magnified $10$ times for clarity
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Figure 3.  Results with potential (22). (a) $2$-Wasserstein distance between the diffusive and plain aggregation solutions for various choices of $\nu$. (b) Energy (4) of solutions to the diffusive model through time for various choices of $\nu$. Also included is the energy (5) of the solution to the particle model through time (solid line). Star markers have been placed at $t = 6.5$, $t = 10.9$ and $t = 16$ for $\nu = 10^{-5}$, $\nu = 10^{-7}$ and $\nu = 10^{-9}$ respectively, corresponding to the times of the first mass transfer
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Figure 4.  Results with potential (22). (a) $2$-Wasserstein distance between solutions to the diffusive model and the (unstable) equilibrium of the plain aggregation model, for various choices of $\nu$. Markers have been placed at $t = 6.5$, $t = 10.9$ and $t = 16$ for $\nu = 10^{-5}$, $\nu = 10^{-7}$ and $\nu = 10^{-9}$ respectively, corresponding to the times of the first mass transfer (see also Figure 3); these times also correspond to when $\mu_\nu(t)$ is closest to $\bar{\mu}$. (b) Solutions to the diffusive model at the times marked in (a). The circles represent concentrations of the equilibrium $\bar{\mu}$, where they have been magnified $25$ times for clarity
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Figure 5.  Results for various values of the exponent $m$. (a) Decreasing $m$ improves the approximation by nonlinear diffusion for fixed $\nu$ (here $\nu = 10^{-5}$). (b) By decreasing $m$, the solutions of the diffusive model pass more closely by the unstable equilibrium of the plain aggregation model
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Figure 6.  Simulations with potential (20) showing early time dynamics. (a) Snapshots of the diffusive model (1) with $\nu = 10^{-6}$. An insert has been included to show the layer of mass near the origin more clearly. (b) Snapshots of the plain aggregation model (2). Concentrations are represented as circle, square, and diamond markers for $t = 0.1$, $t = 0.5$, and $t = 3$ respectively. The masses of concentrations have been magnified 10 times for clarity
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Figure 7.  Results with potential (20). (a) $2$-Wasserstein distance between the diffusive and plain aggregation solutions for various choices of $\nu$. (b) Energy (4) of solutions to the diffusive model through time for various choices of $\nu$. Also included is the energy (5) of the solution to the particle model through time (solid line)
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Figure 8.  Results with potential (20). (a) $2$-Wasserstein distance between solutions to the diffusive model and the (unstable) equilibrium of the plain aggregation model, for various choices of $\nu$. Markers have been placed at $t = 6.9$, $t = 7.6$, and $t = 7.9$ for $\nu = 10^{-4}$, $\nu = 10^{-6}$, and $\nu = 10^{-8}$ respectively, corresponding to the times when $d_W(\mu_\nu(t), \bar{\mu})$ achieves its minimum. (b) Solutions to the diffusive model at the times marked in (a), respectively for each $\nu$. The solid line and the circle marker at origin (indicating a delta concentration) represents the unstable equilibrium $\bar{\mu}$ of the plain aggregation model. The concentration has been magnified $10$ times for clarity
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Figure 9.  (a) Comparison between the energy minimizer $\bar{\mu}^*$ of the plain aggregation model (see (24)) and minimizers ${{\bar \mu }_\nu }$ of the diffusive model (see (30)) for various $\nu$. (b) The $2$-Wasserstein distance between the minimizers ${{\bar \mu }_\nu }$ and $\bar{\mu}^*$ as a function of $\nu$
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Figure 10.  (a) Numerically calculated $c_1$, $c_2$, and $L$ as functions of $\nu$. (b) Numerically calculated behaviour of $c_1 e^{\frac{x}{\sqrt{2\nu}}}$ as $\nu \to 0$ for $x$ at $L$ and near $L$
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Table 1.  $2$-Wasserstein distance $d_W(\mu_\nu(t), \mu(t))$ between solutions of the diffusive model and solutions of the plain aggregation model for various choices of $\nu$ and several early times
$\nu$ $t=0.5$ $t=1$ $t=5$
$10^{-3}$ $2.1400{\rm{e}}-2$ $3.1896{\rm{e}}-2$ $8.6286{\rm{e}}-2$
$10^{-4}$ $7.1776{\rm{e}}-3$ $1.1110{\rm{e}}-2$ $4.7896{\rm{e}}-2$
$10^{-5}$ $3.8652{\rm{e}}-3$ $7.5722{\rm{e}}-3$ $3.3105{\rm{e}}-2$
$10^{-6}$ $3.4619{\rm{e}}-3$ $7.5057{\rm{e}}-3$ $3.3048{\rm{e}}-2$
$10^{-7}$ $3.4555{\rm{e}}-3$ $7.5043{\rm{e}}-3$ $3.2934{\rm{e}}-2$
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$\nu$ $t=0.5$ $t=1$ $t=5$
$10^{-3}$ $2.1400{\rm{e}}-2$ $3.1896{\rm{e}}-2$ $8.6286{\rm{e}}-2$
$10^{-4}$ $7.1776{\rm{e}}-3$ $1.1110{\rm{e}}-2$ $4.7896{\rm{e}}-2$
$10^{-5}$ $3.8652{\rm{e}}-3$ $7.5722{\rm{e}}-3$ $3.3105{\rm{e}}-2$
$10^{-6}$ $3.4619{\rm{e}}-3$ $7.5057{\rm{e}}-3$ $3.3048{\rm{e}}-2$
$10^{-7}$ $3.4555{\rm{e}}-3$ $7.5043{\rm{e}}-3$ $3.2934{\rm{e}}-2$
Table 2.  $2$-Wasserstein distance $d_W(\mu_\nu(t), \mu(t))$ between solutions of the diffusive model and solutions of the plain aggregation model for various choices of $\nu$ at some early times
$\nu$ $t=0.1$ $t=0.5$ $t=3$
$10^{-3}$ $6.8548{\rm{e}}-3$ $2.4142{\rm{e}}-2$ $6.6831{\rm{e}}-2$
$10^{-4}$ $2.9493{\rm{e}}-3$ $1.1424{\rm{e}}-2$ $4.3318{\rm{e}}-2$
$10^{-5}$ $2.3620{\rm{e}}-3$ $8.9352{\rm{e}}-3$ $3.7865{\rm{e}}-2$
$10^{-6}$ $2.3166{\rm{e}}-3$ $8.6382{\rm{e}}-3$ $3.6588{\rm{e}}-2$
$10^{-7}$ $2.3161{\rm{e}}-3$ $8.6054{\rm{e}}-3$ $3.6235{\rm{e}}-2$
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$\nu$ $t=0.1$ $t=0.5$ $t=3$
$10^{-3}$ $6.8548{\rm{e}}-3$ $2.4142{\rm{e}}-2$ $6.6831{\rm{e}}-2$
$10^{-4}$ $2.9493{\rm{e}}-3$ $1.1424{\rm{e}}-2$ $4.3318{\rm{e}}-2$
$10^{-5}$ $2.3620{\rm{e}}-3$ $8.9352{\rm{e}}-3$ $3.7865{\rm{e}}-2$
$10^{-6}$ $2.3166{\rm{e}}-3$ $8.6382{\rm{e}}-3$ $3.6588{\rm{e}}-2$
$10^{-7}$ $2.3161{\rm{e}}-3$ $8.6054{\rm{e}}-3$ $3.6235{\rm{e}}-2$
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