# American Institute of Mathematical Sciences

October  2022, 42(10): 4905-4936. doi: 10.3934/dcds.2022078

## Zero-diffusion limit for aggregation equations over bounded domains

 1 Department of Mathematics, Simon Fraser University, 8888 University Dr, Burnaby, BC V5A 1S6, Canada 2 Department of Mathematics and Statistics, University of Calgary, 2500 University Dr NW, Calgary, AB T2N 1N4, Canada 3 Department of Mathematics, University of Colorado Boulder, Boulder, CO 80309, United States

*Corresponding author: Hui Huang

Received  June 2021 Revised  November 2022 Published  October 2022 Early access  June 2022

Fund Project: RF and WS were supported by NSERC Discovery Grants during this research. HH was partially supported by a postdoctoral fellowship from the Pacific Institute for the Mathematical Sciences (PIMS)

We investigate the zero-diffusion limit for both continuous and discrete aggregation-diffusion models over convex and bounded domains. Our approach relies on a coupling method connecting PDEs with their underlying SDEs. Compared with existing work, our result relaxes the regularity assumptions on the interaction and external potentials and improves the convergence rate (in terms of the diffusion coefficient). The particular rate we derive is shown to be consistent with numerical computations.

Citation: Razvan C. Fetecau, Hui Huang, Daniel Messenger, Weiran Sun. Zero-diffusion limit for aggregation equations over bounded domains. Discrete and Continuous Dynamical Systems, 2022, 42 (10) : 4905-4936. doi: 10.3934/dcds.2022078
##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Springer Science & Business Media, 2008. [2] J. Bedrossian, N. Rodríguez 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] 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. [4] 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. [5] M. Bossy, E. Gobet and D. Talay, A symmetrized Euler scheme for an efficient approximation of reflected diffusions, Journal of Applied Probability, 41 (2004), 877-889.  doi: 10.1239/jap/1091543431. [6] R. Burkard, M. Dell'Amico and S. Martello, Assignment Problems, SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898717754. [7] J. A. Carrillo, K. Craig and F. S. Patacchini, A blob method for diffusion, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 53, 53 pp. doi: 10.1007/s00526-019-1486-3. [8] J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Revista Matematica Iberoamericana, 19 (2003), 971-1018.  doi: 10.4171/RMI/376. [9] J. A. Carrillo, D. Slepčev and L. Wu, Nonlocal-interaction equations on uniformly prox-regular sets, Discrete and Continuous Dynamical Systems-Series A, 36 (2016), 1209-1247.  doi: 10.3934/dcds.2016.36.1209. [10] L. Chen, S. Göttlich and S. Knapp, Modeling of a diffusion with aggregation: Rigorous derivation and numerical simulation, ESAIM Math. Model. Numer. Anal., 52 (2018), 567-593.  doi: 10.1051/m2an/2018028. [11] Y.-P. Choi and S. Salem, Propagation of chaos for aggregation equations with no-flux boundary conditions and sharp sensing zones, Mathematical Models and Methods in Applied Sciences, 28 (2018), 223-258.  doi: 10.1142/S0218202518500070. [12] J. H. M. Evers and T. Kolokolnikov, Metastable states for an aggregation model with noise, SIAM Journal on Applied Dynamical Systems, 15 (2016), 2213-2226.  doi: 10.1137/16M1069006. [13] 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. [14] R. C. Fetecau, H. Huang and W. Sun, Propagation of chaos for the Keller-Segel equation over bounded domains, Journal of Differential Equations, 266 (2019), 2142-2174.  doi: 10.1016/j.jde.2018.08.024. [15] 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. [16] 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. [17] R. C. Fetecau, M. Kovacic and I. Topaloglu, Swarming in domains with boundaries: Approximation and regularization by nonlinear diffusion, Discrete and Continuous Dynamical Systems - Series B, 24 (2019), 1815-1842.  doi: 10.3934/dcdsb.2018238. [18] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides: Control Systems, vol. 18, Springer Science & Business Media, 2013. doi: 10.1007/978-94-015-7793-9. [19] N. Fournier and A. Guillin, On the rate of convergence in Wasserstein distance of the empirical measure, Probability Theory and Related Fields, 162 (2015), 707-738.  doi: 10.1007/s00440-014-0583-7. [20] N. Fournier, M. Hauray and S. Mischler, Propagation of chaos for the 2d viscous vortex model, Journal of the European Mathematical Society, 16 (2014), 1423-1466.  doi: 10.4171/JEMS/465. [21] V. Gazi and K. M. Passino, Stability analysis of swarms, IEEE Transactions on Automatic Control, 48 (2003), 692-697.  doi: 10.1109/TAC.2003.809765. [22] I. I. Gihman and A. V. Skorohod, Stochastic Differential Equations, Springer-Verlag Berlin Heidelberg, New York, 1972. [23] D. D. Holm and V. Putkaradze, Formation of clumps and patches in self-aggregation of finite-size particles, Physica D: Nonlinear Phenomena, 220 (2006), 183-196.  doi: 10.1016/j.physd.2006.07.010. [24] H. Huang and J.-G. Liu, Discrete-in-time random particle blob method for the Keller-Segel equation and convergence analysis, Communication in Mathematical Sciences, 15 (2017), 1821-1842.  doi: 10.4310/CMS.2017.v15.n7.a2. [25] H. Huang and J.-G. Liu, Error estimate of a random particle blob method for the Keller-Segel equation, Mathematics of Computation, 86 (2017), 2719-2744.  doi: 10.1090/mcom/3174. [26] H. Huang, J.-G. Liu and J. Lu, Learning interacting particle systems: Diffusion parameter estimation for aggregation equations, Mathematical Models and Methods in Applied Sciences, 29 (2019), 1-29.  doi: 10.1142/S0218202519500015. [27] P.-E. Jabin and Z. Wang, Mean field limit for stochastic particle systems, in Active Particles, Volume 1, Springer, 2017,379–402. [28] P.-E. Jabin and Z. Wang, Quantitative estimates of propagation of chaos for stochastic systems with ${W}^{-1, \infty}$ kernels, Inventiones Mathematicae, 214 (2018), 523-591.  doi: 10.1007/s00222-018-0808-y. [29] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [30] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1995. doi: 10.1007/978-3-662-12616-5. [31] 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). doi: 10.1103/PhysRevE.84.015203. [32] P.-L. Lions and A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Communications on Pure and Applied Mathematics, 37 (1984), 511-537.  doi: 10.1002/cpa.3160370408. [33] D. Messenger and R. C. Fetecau, Equilibria of an aggregation model with linear diffusion in domains with boundaries, Mathematical Models and Methods in Applied Sciences, 30 (2020), 805-845.  doi: 10.1142/S0218202520400059. [34] S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577–621. doi: 10.1137/120901866. [35] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Sixth Edition, Springer, Berlin, 2003. doi: 10.1007/978-3-642-14394-6. [36] Y. Saisho, Stochastic differential equations for multi-dimensional domain with reflecting boundary, Probability Theory and Related Fields, 74 (1987), 455-477.  doi: 10.1007/BF00699100. [37] S. Serfaty, Mean field limit for coulomb-type flows, Duke Mathematical Journal, 169 (2020), 2887-2935.  doi: 10.1215/00127094-2020-0019. [38] A. V. Skorokhod, Stochastic equations for diffusion processes in a bounded region, Theory of Probability & Its Applications, 6 (1961), 264-274.  doi: 10.1137/1106035. [39] A. V. Skorokhod, Stochastic equations for diffusion processes in a bounded region. Ⅱ, Theory of Probability & Its Applications, 7 (1962), 3-23.  doi: 10.1137/1107002. [40] A.-S. Sznitman, Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated, Journal of Functional Analysis, 56 (1984), 311-336.  doi: 10.1016/0022-1236(84)90080-6. [41] H. Tanaka, Stochastic differential equations with reflecting, Stochastic Processes: Selected Papers of Hiroshi Tanaka, 9 (1979), 157. [42] 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, 11 pp. doi: 10.1371/journal.pcbi.1002642. [43] C. Villani, Optimal Transport: Old and New, vol. 338, Springer Science & Business Media, 2008. doi: 10.1007/978-3-540-71050-9. [44] E. Weinan, Dynamics of vortex liquids in Ginzburg-Landau theories with applications to superconductivity, Physical Review B, 50 (1994), 1126.  doi: 10.1103/PhysRevB.50.1126. [45] L. Wu and D. Slepčev, Nonlocal interaction equations in environments with heterogeneities and boundaries, Communications in Partial Differential Equations, 40 (2015), 1241-1281.  doi: 10.1080/03605302.2015.1015033. [46] Y. Zhang, On continuity equations in space-time domains, Discrete and Continuous Dynamical Systems-Series A, 38 (2018), 4837-4873.  doi: 10.3934/dcds.2018212.

show all references

##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Springer Science & Business Media, 2008. [2] J. Bedrossian, N. Rodríguez 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] 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. [4] 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. [5] M. Bossy, E. Gobet and D. Talay, A symmetrized Euler scheme for an efficient approximation of reflected diffusions, Journal of Applied Probability, 41 (2004), 877-889.  doi: 10.1239/jap/1091543431. [6] R. Burkard, M. Dell'Amico and S. Martello, Assignment Problems, SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898717754. [7] J. A. Carrillo, K. Craig and F. S. Patacchini, A blob method for diffusion, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 53, 53 pp. doi: 10.1007/s00526-019-1486-3. [8] J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Revista Matematica Iberoamericana, 19 (2003), 971-1018.  doi: 10.4171/RMI/376. [9] J. A. Carrillo, D. Slepčev and L. Wu, Nonlocal-interaction equations on uniformly prox-regular sets, Discrete and Continuous Dynamical Systems-Series A, 36 (2016), 1209-1247.  doi: 10.3934/dcds.2016.36.1209. [10] L. Chen, S. Göttlich and S. Knapp, Modeling of a diffusion with aggregation: Rigorous derivation and numerical simulation, ESAIM Math. Model. Numer. Anal., 52 (2018), 567-593.  doi: 10.1051/m2an/2018028. [11] Y.-P. Choi and S. Salem, Propagation of chaos for aggregation equations with no-flux boundary conditions and sharp sensing zones, Mathematical Models and Methods in Applied Sciences, 28 (2018), 223-258.  doi: 10.1142/S0218202518500070. [12] J. H. M. Evers and T. Kolokolnikov, Metastable states for an aggregation model with noise, SIAM Journal on Applied Dynamical Systems, 15 (2016), 2213-2226.  doi: 10.1137/16M1069006. [13] 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. [14] R. C. Fetecau, H. Huang and W. Sun, Propagation of chaos for the Keller-Segel equation over bounded domains, Journal of Differential Equations, 266 (2019), 2142-2174.  doi: 10.1016/j.jde.2018.08.024. [15] 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. [16] 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. [17] R. C. Fetecau, M. Kovacic and I. Topaloglu, Swarming in domains with boundaries: Approximation and regularization by nonlinear diffusion, Discrete and Continuous Dynamical Systems - Series B, 24 (2019), 1815-1842.  doi: 10.3934/dcdsb.2018238. [18] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides: Control Systems, vol. 18, Springer Science & Business Media, 2013. doi: 10.1007/978-94-015-7793-9. [19] N. Fournier and A. Guillin, On the rate of convergence in Wasserstein distance of the empirical measure, Probability Theory and Related Fields, 162 (2015), 707-738.  doi: 10.1007/s00440-014-0583-7. [20] N. Fournier, M. Hauray and S. Mischler, Propagation of chaos for the 2d viscous vortex model, Journal of the European Mathematical Society, 16 (2014), 1423-1466.  doi: 10.4171/JEMS/465. [21] V. Gazi and K. M. Passino, Stability analysis of swarms, IEEE Transactions on Automatic Control, 48 (2003), 692-697.  doi: 10.1109/TAC.2003.809765. [22] I. I. Gihman and A. V. Skorohod, Stochastic Differential Equations, Springer-Verlag Berlin Heidelberg, New York, 1972. [23] D. D. Holm and V. Putkaradze, Formation of clumps and patches in self-aggregation of finite-size particles, Physica D: Nonlinear Phenomena, 220 (2006), 183-196.  doi: 10.1016/j.physd.2006.07.010. [24] H. Huang and J.-G. Liu, Discrete-in-time random particle blob method for the Keller-Segel equation and convergence analysis, Communication in Mathematical Sciences, 15 (2017), 1821-1842.  doi: 10.4310/CMS.2017.v15.n7.a2. [25] H. Huang and J.-G. Liu, Error estimate of a random particle blob method for the Keller-Segel equation, Mathematics of Computation, 86 (2017), 2719-2744.  doi: 10.1090/mcom/3174. [26] H. Huang, J.-G. Liu and J. Lu, Learning interacting particle systems: Diffusion parameter estimation for aggregation equations, Mathematical Models and Methods in Applied Sciences, 29 (2019), 1-29.  doi: 10.1142/S0218202519500015. [27] P.-E. Jabin and Z. Wang, Mean field limit for stochastic particle systems, in Active Particles, Volume 1, Springer, 2017,379–402. [28] P.-E. Jabin and Z. Wang, Quantitative estimates of propagation of chaos for stochastic systems with ${W}^{-1, \infty}$ kernels, Inventiones Mathematicae, 214 (2018), 523-591.  doi: 10.1007/s00222-018-0808-y. [29] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [30] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1995. doi: 10.1007/978-3-662-12616-5. [31] 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). doi: 10.1103/PhysRevE.84.015203. [32] P.-L. Lions and A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Communications on Pure and Applied Mathematics, 37 (1984), 511-537.  doi: 10.1002/cpa.3160370408. [33] D. Messenger and R. C. Fetecau, Equilibria of an aggregation model with linear diffusion in domains with boundaries, Mathematical Models and Methods in Applied Sciences, 30 (2020), 805-845.  doi: 10.1142/S0218202520400059. [34] S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577–621. doi: 10.1137/120901866. [35] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Sixth Edition, Springer, Berlin, 2003. doi: 10.1007/978-3-642-14394-6. [36] Y. Saisho, Stochastic differential equations for multi-dimensional domain with reflecting boundary, Probability Theory and Related Fields, 74 (1987), 455-477.  doi: 10.1007/BF00699100. [37] S. Serfaty, Mean field limit for coulomb-type flows, Duke Mathematical Journal, 169 (2020), 2887-2935.  doi: 10.1215/00127094-2020-0019. [38] A. V. Skorokhod, Stochastic equations for diffusion processes in a bounded region, Theory of Probability & Its Applications, 6 (1961), 264-274.  doi: 10.1137/1106035. [39] A. V. Skorokhod, Stochastic equations for diffusion processes in a bounded region. Ⅱ, Theory of Probability & Its Applications, 7 (1962), 3-23.  doi: 10.1137/1107002. [40] A.-S. Sznitman, Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated, Journal of Functional Analysis, 56 (1984), 311-336.  doi: 10.1016/0022-1236(84)90080-6. [41] H. Tanaka, Stochastic differential equations with reflecting, Stochastic Processes: Selected Papers of Hiroshi Tanaka, 9 (1979), 157. [42] 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, 11 pp. doi: 10.1371/journal.pcbi.1002642. [43] C. Villani, Optimal Transport: Old and New, vol. 338, Springer Science & Business Media, 2008. doi: 10.1007/978-3-540-71050-9. [44] E. Weinan, Dynamics of vortex liquids in Ginzburg-Landau theories with applications to superconductivity, Physical Review B, 50 (1994), 1126.  doi: 10.1103/PhysRevB.50.1126. [45] L. Wu and D. Slepčev, Nonlocal interaction equations in environments with heterogeneities and boundaries, Communications in Partial Differential Equations, 40 (2015), 1241-1281.  doi: 10.1080/03605302.2015.1015033. [46] Y. Zhang, On continuity equations in space-time domains, Discrete and Continuous Dynamical Systems-Series A, 38 (2018), 4837-4873.  doi: 10.3934/dcds.2018212.
Illustration of the projection ${\pi}_{_{\partial D}}$ (see (94)) and the reflection operator ${\bf{R}}$ (see (93)) for a point $x \notin D$
(Top) Convergence results for $D_H = [0, \infty)\times {\mathbb R}$ with uniform initial distribution on $[0, 0.25]\times[-0.125, 0.125]$. (Bottom) Convergence results for $D_C = \{|x| \leq 0.2\}$ with uniform initial distribution on $[-0.05, 0.05]\times[0, 0.1]$. Left: $K_2$, right: $K_{3/2}$. Agreement with the theoretical rate of $\mathcal{O}(\nu)$ is very strong up to times near $T = 1$, after which (for reasons discussed in the text) convergence at the expected rate is seen asymptotically, below a threshold value of $\nu$
Evolution of $E^\nu({t})$ over time with $\nu = 2^{-28} \approx 10^{-8.4}$ and interaction potential $K_{3/2}$ for the half-plane $D_H$ (left) and disk $D_C$ (right). As expected, variance of $E^\nu({t})$ grows significantly over time (as seen by the 95% confidence intervals), yet we see excellent agreement using a nonlinear least-squares fit to a curve of the form $y(t) = at(1+bte^{bt})$, matching that of the bounding curve in Theorem 4.4
Swarming in the half-plane $D_H$ with the interaction potential $K_{3/2}$; here $\Delta t = 2^{-8}$ and $\nu = 2^{-16}$. Circles and filled-in squares represent non-diffusive and diffusive particles, respectively. Particles rapidly expand until $T\approx 1$, after which attractive forces confine the swarm. At $T = 1$ a representative particle is circled in the space between the boundary aggregation and the free swarm, illustrating the pair-separation effect mentioned in the text. By time $T = 100$ the swarm has nearly escaped the boundary, forming a disk in free space, with diffusive particles forming concentrated clumps and non-diffusive particles forming $\delta$-aggregations
Swarming in the disk $D_C$ with $K, \nu$ an $\Delta t$ matching that of Figure 4. By time $T\approx 1$, particles have formed a boundary aggregation and a free swarm as in domain $D_H$, with another representative particle (circled) caught between the boundary and free swarm. By time $T = 100$ the swarm has formed a disk of particle aggregates centred in the domain along with periodic aggregations along the boundary
Weak convergence of the Euler scheme with symmetric reflection (95), with diffusion coefficient $\nu = 0.01$, $N = 5$ particles, domain $D_C$ and interaction potential $K_{3/2}$. We use the test function $g(\tilde{X}_{\tau_{{n}}}^{\nu}) = \frac{1}{N}\sum_{i = 1}^N\left\vert \tilde{X}_{\tau_{{n}}}^{\nu, {i}}\right\vert^2$ which is motivated by the computation of (97) as discussed in the text. In lieu of an exact solution we compare at time $\tau_n = 0.25$ to a "fine-grid" solution $X^{fine}_{\tau_n}$ computed with timestep $\Delta t = 2^{-15}$. To approximate expectations we use $2\times 10^5$ samples of Brownian motion, each simulation sharing the same initial data. Left: semi-log plot showing convergence of $\mathbb{E}\left[g\left(\tilde{X}_{\tau_{{n}}}^{\nu}\right)\right]$ to $\mathbb{E}\left[g\left(X^{fine}_{\tau_n}\right)\right]$ as well as 95% confidence intervals. Right: log-log plot showing convergence rate at approximately $\mathcal{O}(\Delta t)$
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