# American Institute of Mathematical Sciences

2012, 17(4): 1309-1331. doi: 10.3934/dcdsb.2012.17.1309

## Asymptotics of blowup solutions for the aggregation equation

 1 Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC V5A 1S6, Canada 2 520 Portola Plaza, Math Sciences Building 6363, Los Angeles, CA 90095

Received  January 2011 Revised  September 2011 Published  February 2012

We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation $u_t = \nabla\cdot(u\nabla K*u)$ in $\mathbb{R}^n$, for homogeneous potentials $K(x) = |x|^\gamma$, $\gamma>0$. For $\gamma>2$, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing $\delta$-ring. We develop an asymptotic theory for the approach to this singular solution. For $\gamma < 2$, the solution blows up in finite time and we present careful numerics of second type similarity solutions for all $\gamma$ in this range, including additional asymptotic behaviors in the limits $\gamma \to 0^+$ and $\gamma\to 2^-$.
Citation: Yanghong Huang, Andrea Bertozzi. Asymptotics of blowup solutions for the aggregation equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1309-1331. doi: 10.3934/dcdsb.2012.17.1309
##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,", Second edition, (2008). [2] D. G. Aronson and J. L. Vázquez, Anomalous exponents in nonlinear diffusion,, J. Nonlinear Sci., 5 (1995), 29. [3] D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media,, RAIRO Modél. Math. Anal. Numér., 31 (1997), 615. [4] A. L. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation,, Commun. Math. Sci., 8 (2010), 45. [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. doi: 10.1088/0951-7715/22/3/009. [6] A. L. Bertozzi, J. B. Garnett and T. Laurent, Characterization of radially symmetric finite time blowup in multidimensional aggregation equations,, to appear in SIAM J. Math. Anal., (2012). [7] A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in $\mathbf R^n$,, Comm. Math. Phys., 274 (2007), 717. doi: 10.1007/s00220-007-0288-1. [8] A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation,, Comm. Pur. Appl. Math., 64 (2011), 45. doi: 10.1002/cpa.20334. [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. doi: 10.1016/j.jde.2005.07.025. [10] M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions,, Nonlinear Anal. Real World Appl., 8 (2007), 939. doi: 10.1016/j.nonrwa.2006.04.002. [11] M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion,, Networks and Heterogeneous Media, 3 (2008), 749. doi: 10.3934/nhm.2008.3.749. [12] E. Caglioti and C. Villani, Homogeneous cooling states are not always good approximations to granular flows,, Arch. Ration. Mech. Anal., 163 (2002), 329. doi: 10.1007/s002050200204. [13] J. A. Carrillo, M. DiFrancesco, 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. doi: 10.1215/00127094-2010-211. [14] Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system,, Phys. D, 232 (2007), 33. doi: 10.1016/j.physd.2007.05.007. [15] H. Dong, The aggregation equation with power-law kernels: ill-posedness, mass concentration and similarity solutions,, Communications in Mathematical Physics, 304 (2011), 649. doi: 10.1007/s00220-011-1237-6. [16] M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and Lincoln S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse,, Phys. Rev. Lett., 96 (2006). [17] J. Eggers and M. A. Fontelos, The role of self-similarity in singularities of partial differential equations,, Nonlinearity, 22 (2009). doi: 10.1088/0951-7715/22/1/R01. [18] D. D. Holm and V. Putkaradze, Aggregation of finite-size particles with variable mobility,, Phys. Rev. Lett., 95 (2005). doi: 10.1103/PhysRevLett.95.226106. [19] D. D. Holm and V. Putkaradze, Formation of clumps and patches in self-aggregation of finite-size particles,, Phys. D, 220 (2006), 183. doi: 10.1016/j.physd.2006.07.010. [20] Y. Huang, "Self-Similar Blowup Solutions of the Aggregation Equation,'', Ph.D thesis, (2010), 10. [21] Y. Huang and A. L. Bertozzi, Self-similar blowup solutions to an aggregation equation in $R^n$,, SIAM J. Appl. Math., 70 (2010), 2582. doi: 10.1137/090774495. [22] Y. Huang, T. Witelski and A. L. Bertozzi, Anomalous exponents of self-similar blowup solution to an aggregation equation in odd dimensions,, preprint, (2010). [23] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. [24] T. Laurent, Local and global existence for an aggregation equation,, Comm. Partial Differential Equations, 32 (2007), 1941. [25] H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows,, Arch. Ration. Mech. Anal., 172 (2004), 407. doi: 10.1007/s00205-004-0307-8. [26] A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm,, J. Math. Biol., 38 (1999), 534. doi: 10.1007/s002850050158. [27] A. Mogilner, L. Edelstein-Keshet, L. Bent and A. Spiros, Mutual interactions, potentials, and individual distance in a social aggregation,, J. Math. Biol., 47 (2003), 353. doi: 10.1007/s00285-003-0209-7. [28] D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations,, J. Math. Biol., 50 (2005), 49. doi: 10.1007/s00285-004-0279-1. [29] C. M. Topaz, A. J. Bernoff, S. Logan and W. Toolson, A model for rolling swarms of locusts,, The European Physical Journal-Special Topics, 157 (2008), 93. doi: 10.1140/epjst/e2008-00633-y. [30] C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups,, SIAM J. Appl. Math., 65 (2004), 152. doi: 10.1137/S0036139903437424. [31] C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation,, Bull. Math. Biol., 68 (2006), 1601. doi: 10.1007/s11538-006-9088-6. [32] G. Toscani, One-dimensional kinetic models of granular flows,, M2AN Math. Model. Numer. Anal., 34 (2000), 1277. doi: 10.1051/m2an:2000127. [33] C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58,, American Mathematical Society, (2003).

show all references

##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,", Second edition, (2008). [2] D. G. Aronson and J. L. Vázquez, Anomalous exponents in nonlinear diffusion,, J. Nonlinear Sci., 5 (1995), 29. [3] D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media,, RAIRO Modél. Math. Anal. Numér., 31 (1997), 615. [4] A. L. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation,, Commun. Math. Sci., 8 (2010), 45. [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. doi: 10.1088/0951-7715/22/3/009. [6] A. L. Bertozzi, J. B. Garnett and T. Laurent, Characterization of radially symmetric finite time blowup in multidimensional aggregation equations,, to appear in SIAM J. Math. Anal., (2012). [7] A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in $\mathbf R^n$,, Comm. Math. Phys., 274 (2007), 717. doi: 10.1007/s00220-007-0288-1. [8] A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation,, Comm. Pur. Appl. Math., 64 (2011), 45. doi: 10.1002/cpa.20334. [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. doi: 10.1016/j.jde.2005.07.025. [10] M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions,, Nonlinear Anal. Real World Appl., 8 (2007), 939. doi: 10.1016/j.nonrwa.2006.04.002. [11] M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion,, Networks and Heterogeneous Media, 3 (2008), 749. doi: 10.3934/nhm.2008.3.749. [12] E. Caglioti and C. Villani, Homogeneous cooling states are not always good approximations to granular flows,, Arch. Ration. Mech. Anal., 163 (2002), 329. doi: 10.1007/s002050200204. [13] J. A. Carrillo, M. DiFrancesco, 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. doi: 10.1215/00127094-2010-211. [14] Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system,, Phys. D, 232 (2007), 33. doi: 10.1016/j.physd.2007.05.007. [15] H. Dong, The aggregation equation with power-law kernels: ill-posedness, mass concentration and similarity solutions,, Communications in Mathematical Physics, 304 (2011), 649. doi: 10.1007/s00220-011-1237-6. [16] M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and Lincoln S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse,, Phys. Rev. Lett., 96 (2006). [17] J. Eggers and M. A. Fontelos, The role of self-similarity in singularities of partial differential equations,, Nonlinearity, 22 (2009). doi: 10.1088/0951-7715/22/1/R01. [18] D. D. Holm and V. Putkaradze, Aggregation of finite-size particles with variable mobility,, Phys. Rev. Lett., 95 (2005). doi: 10.1103/PhysRevLett.95.226106. [19] D. D. Holm and V. Putkaradze, Formation of clumps and patches in self-aggregation of finite-size particles,, Phys. D, 220 (2006), 183. doi: 10.1016/j.physd.2006.07.010. [20] Y. Huang, "Self-Similar Blowup Solutions of the Aggregation Equation,'', Ph.D thesis, (2010), 10. [21] Y. Huang and A. L. Bertozzi, Self-similar blowup solutions to an aggregation equation in $R^n$,, SIAM J. Appl. Math., 70 (2010), 2582. doi: 10.1137/090774495. [22] Y. Huang, T. Witelski and A. L. Bertozzi, Anomalous exponents of self-similar blowup solution to an aggregation equation in odd dimensions,, preprint, (2010). [23] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. [24] T. Laurent, Local and global existence for an aggregation equation,, Comm. Partial Differential Equations, 32 (2007), 1941. [25] H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows,, Arch. Ration. Mech. Anal., 172 (2004), 407. doi: 10.1007/s00205-004-0307-8. [26] A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm,, J. Math. Biol., 38 (1999), 534. doi: 10.1007/s002850050158. [27] A. Mogilner, L. Edelstein-Keshet, L. Bent and A. Spiros, Mutual interactions, potentials, and individual distance in a social aggregation,, J. Math. Biol., 47 (2003), 353. doi: 10.1007/s00285-003-0209-7. [28] D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations,, J. Math. Biol., 50 (2005), 49. doi: 10.1007/s00285-004-0279-1. [29] C. M. Topaz, A. J. Bernoff, S. Logan and W. Toolson, A model for rolling swarms of locusts,, The European Physical Journal-Special Topics, 157 (2008), 93. doi: 10.1140/epjst/e2008-00633-y. [30] C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups,, SIAM J. Appl. Math., 65 (2004), 152. doi: 10.1137/S0036139903437424. [31] C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation,, Bull. Math. Biol., 68 (2006), 1601. doi: 10.1007/s11538-006-9088-6. [32] G. Toscani, One-dimensional kinetic models of granular flows,, M2AN Math. Model. Numer. Anal., 34 (2000), 1277. doi: 10.1051/m2an:2000127. [33] C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58,, American Mathematical Society, (2003).
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