Characterization of radially symmetric finite time blowup in multidimensional aggregation equations

SIAM Journal on Mathematical Analysis

Volumn 44, Issue 2, 2012, Pages 651-681

  Bertozzi, Andrea L ^a   Garnett, John B ^a   Laurent, Thomas ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Blowup; Measure solutions; Nonlocal partial differential equation

Indexed keywords

BLOW-UP PROFILE; BLOW-UP TIME; BLOWUP; BURGERS EQUATIONS; CLASSICAL THEORY; CONSERVATION LAW; CRITICAL EXPONENT; FINITE TIME; FINITE TIME BLOW-UP; HALF-LINE; ILL-POSEDNESS; MEASURE SOLUTIONS; MONOTONICITY; NEWTONIANS; NONLOCAL; NUMERICAL STUDIES; RADIAL SYMMETRYS; RADIALLY SYMMETRIC; RADIALLY SYMMETRIC SOLUTION;

MATHEMATICAL TECHNIQUES;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 84861370911     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/11081986X     Document Type: Article

References (77)

1. L.A. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures Math. ETH Zürich, Birkhäuser, Basel, Switzerland, 2005.
2. L. Ambrosio, E. Mainini, and S. Serfaty, Gradient Flow of the Chapman-Rubinstein-Schatzman Model for Signed Vortices. Ann. Inst. H. Poincaré AnalNon Linéaire, 28 (2011), pp. 217-246.
3. L. Ambrosio and S. Serfaty, A gradient flow approach to an evolution problem arising in superconductivity. Commun. Pure Appl. Math., 16 (2008), pp. 1495-1539.
4. C. Bardos, A.Y. Leroux, and J.C. Nedelec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), pp. 1017-1034.
5. J. Bedrossian, N. Rodriguez, and A.L. Bertozzi, Local and global well-posedness for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion, Nonlinearity, 24 (2011), pp. 1683-1714.
6. J.D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), pp. 375-393.
7. D. Benedetto, E. Caglioti, and M. Pulvirenti, A kinetic equation for granular media, RAIRO Modél. Math. Anal. Numér., 31, (1997), pp. 615-641. (Pubitemid 127374202)
8. A.L. Bertozzi and J. Brandman, Finite-time blow-up of L8-weak solutions of an aggregation equation, Commun. Math. Sci., 8 (2010), pp. 45-65.
9. A.L. Bertozzi, J.A. Carrillo, and T. Laurent, Blowup in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), pp. 683-710.
10. A.L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in Rn, Comm. Math. Phys., 274 (2007), pp. 717-735.
11. A.L. Bertozzi, T. Laurent, and F. Leger, Aggregation and spreading via the Newtonian potential: The dynamics of patch solutions, M3AS, to appear.
12. A.L. Bertozzi, T. Laurent, and J. Rosado, Lp Theory for the aggregation equation, Commun. Pure Appl. Math., 64 (2011), pp. 45-83.
13. A.L. Bertozzi and D. Slepcev, Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion, Commun. Pure Appl. Anal., 9, (2010), pp. 1617-1637.
14. P. Biler, Growth and accretion of mass in an astrophysical model, Appl. Math., 23 (1995), pp. 179-189.
15. P. Biler and A. Woyczyński, Global and expoding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math., 59 (1998), pp. 845-869.
16. A. Blanchet, J.A. Carrillo, and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in R2, Commun. Pure Appl. Math., 61 (2008), pp. 1449-1481.
17. A. Blanchet, J. Dolbeault, and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 44 (2006).
18. M. Bodnar and J.J.L. Velázquez, An integro-differential equation arising as a limit of individual cell-based models, J. Differential Equations, 222 (2006), pp. 341-380. (Pubitemid 43246216)
19. S. Boi, V. Capasso, and D. Morale, Modeling the aggregative behavior of ants of the species Polyergus rufescens, Nonlinear Anal. Real World Appl., 1 (2000), pp. 163-176.
20. Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Commun. Pure Appl. Math., 44 (1991), pp. 375-417.
21. M.P. Brenner, P. Constantin, L.P. Kadanoff, A. Schenkel, and S.C. Venkataramani, Diffusion, attraction and collapse, Nonlinearity, 12 (1999), pp. 1071-1098.
22. M. Burger, V. Capasso, and D. Morale, On an aggregation model with long and short range interactions, Nonlinear Anal. Real World Appl., 8 (2007), pp. 939-958. (Pubitemid 46097518)
23. M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Netw. Heterogenous Media, 3 (2008), pp. 749-785.
24. J.A. Carrillo, M. Difrancesco, A. Figalli, T. Laurent, and D. Slepčev, Global-intime weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), pp. 229-271.
25. J.A. Carrillo, R.J. McCann, and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19 (2003), pp. 1-48.
26. 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), pp. 217-263. (Pubitemid 43237778)
27. C. Chicone, Ordinary Differential Equations with Applications, Texts in Appl. Math., Springer, Berlin, 1999.
28. J.A. Canizo, J.A. Carrillo, and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), pp. 515-539.
29. J.A. Carrillo and J. Rosado, Uniqueness of bounded solutions to aggregation equations by optimal transport methods, in Proceedings of the 5th European Congress of Mathematicians, 2010, pp. 3-16.
30. J.-M. Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc., 4 (1991), pp. 553-586.
31. R. Diperna and A.J. Majda, Reduced Hausdorff dimension and concentration-cancellation for two dimensional incompressible flow, J. Amer. Math. Soc., 1 (1988), pp. 59-95.
32. J. Dolbeault and B. Perthame, Optimal critical mass in the two-dimensional Keller-Segel model in R2, C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 611-616. (Pubitemid 41447527)
33. H. Dong, The aggregation equation with power-law kernels: Ill-posedness, mass concentration and similarity solutions, Commun. Math. Phys., 304 (2011), pp. 649-664.
34. H. Dong, On similarity solutions to the multidimensional aggregation equation, SIAM J.Math. Anal., 43 (2011), pp. 1995-2008.
35. Q. Du and P. Zhang, Existence of weak solutions to some vortex density models, SIAM J. Math. Anal., 34 (2003), pp. 1279-1299. (Pubitemid 37416689)
36. E. Weinan, Dynamics of vortex liquids in Ginzburg-Landau theories with applications to superconductivity, Phys. Rev. B, 50 (1994), pp. 1126-1135.
37. L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.
38. W. Gangbo and R.J. McCann, The geometry of optimal transportation, Acta Math., 177 (1996), pp. 113-161. (Pubitemid 126345872)
39. V. Gazi and K. Passino, Stability analysis of swarms, IEEE Trans. Automat. Control, 48 (2003), pp. 692-697.
40. P. Hartman, Ordinary Differential Equations, 2nd ed, Birkhäuser, Boston, 1982.
41. D. Holm and V. Putkaradze, Formation of clumps and patches in self-aggregation of finitesize particles, Phys. D, 220 (2006), pp. 183-196. (Pubitemid 44216019)
42. D. Holm and V. Putkaradze, Aggregation of finite size particles with variable mobility, Phys. Rev. Lett., 95 (2005), 226106.
43. Y. Huang, Self-Similar Blowup Solutions of the Aggregation Equation, Ph.D. thesis, University of California Los Angeles, 2010, http://www.math.ucla. edu/applied/cam/.
44. Y. Huang and A.L. Bertozzi, Self-similar blow-up solutions to an aggregation equation in Rn, SIAM J. Appl. Math., 70 (2010), pp. 2582-2603.
45. Y. Huang and A.L. Bertozzi, Asymptotics of blowup solutions for the aggregation equation, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), pp. 1309-1331.
46. E.F. Keller and L.A. Segel, Initiation of slide mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970).
47. T. Kolokolnikov, H. Sun, D. Uminsky, and A.L. Bertozzi, Stability of ring patterns arising from 2D particle interactions, Phys. Rev. E Rapid Commun., 84 (2011), 015203(R).
48. T. Laurent, Local and global existence for an aggregation equation, Comm. Partial Differential Equations, 32 (2007), pp. 1941-1964.
49. P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS-NSF Regional Conf. Ser. in Appl. Math. 11, SIAM, Philadelphia, 1973.
50. D. Li and J. Rodrigo, Finite-time singularities of an aggregation equation in Rn with fractional dissipation, Commun. Math. Phys., 287 (2009), pp. 687-703.
51. D. Li and J. Rodrigo, Refined blowup criteria and nonsymmetric blowup of an aggregation equation, Adv. Math., 220 (2009), pp. 1717-1738.
52. H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Arch. Ration. Mech. Anal., 172 (2004), pp. 407-428. (Pubitemid 38808952)
53. D. Li and X. Zhang, On a nonlocal aggregation model with nonlinear diffusion, Contin. Dyn. Syst. Ser. A, 27 (2010), pp. 301-323.
54. E. Lieb, Sharp constant in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), pp. 349-374.
55. F. Lin and P. Zhang On the hydrodynamic limit of Ginzburg-Landau vortices, Discrete Contin. Dyn. Syst., 6 (2000), pp. 121-142.
56. E. Mainini, A global uniqueness result for an evolution problem arising in superconductivity, Boll. Unione Mat. Ital., 2009, pp. 509-528.
57. A.J. Majda and A.L. Bertozzi Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, UK, 2001.
58. N. Masmoudi and P. Zhang, Global solutions to vortex density equations arising from supconductivity, Ann. Inst. H. Poincaré, 22 (2005), pp. 441-458. (Pubitemid 40691117)
59. G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl., 86 (2006), pp. 68-79. (Pubitemid 44009822)
60. M.C. Lopes Filho, H.J. Nussenzveig Lopes, and S. Schochet, A criterion for the equivalence of the Birkhoff-Rott and Euler descriptions of vortex sheet evolution, Trans. Amer. Math. Soc., 359 (2007), pp. 4125-4142.
61. R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), pp. 153-179. (Pubitemid 127171227)
62. A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), pp. 534-570.
63. D. Morale, V. Capasso, and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), pp. 49-66. (Pubitemid 40240753)
64. J. Nieto, F. Poupaud, and J. Soler, High-field limit for the Vlasov-Poisson-Fokker-Planck system, Arch. Ration. Mech. Anal., 158 (2001), pp. 29-59. (Pubitemid 33577049)
65. F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal., 9 (2002), pp. 533-561.
66. E. Sandier and S. Serfaty, A rigorous derivation of a free-boundary problem arising in superconductivity, Ann. Sci. École Norm. Sup., 33 (2000), pp. 561-592.
67. E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg-Landau Model, Progr. Nonlinear Differential Equations Appl., 70, Birkhäuser, Boston, 2007.
68. H. Sun, D. Uminsky, and A.L. Bertozzi, A generalized Birkhoff-Rott equation for 2D active scalar problems, SIAM J. Appl. Math., 72 (2012), pp. 382-404.
69. C.M. Topaz and A.L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), pp. 152-174.
70. C.M. Topaz, A.L. Bertozzi, and M.A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), pp. 1601-1623. (Pubitemid 44392962)
71. G. Toscani, One-dimensional kinetic models of granular flows, RAIRO Modél. Math. Anal. Numér., 34 (2000), pp. 1277-1291. (Pubitemid 33220541)
72. J.L. Vazquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations-Equations of Porous Medium Type, Oxford University Press, Oxford, UK, 2006.
73. C. Villani, Topics in Optimal Transportation, Grad. Studies in Math. 58, American Mathematical Society, Providence, RI, 2003.
74. C. Villani, Optimal Transport, Old and New, Grundlehren der mathematischen Wissenschaften, Springer, 2009.
75. J. von Brecht and A.L. Bertozzi, Well-posedness theory for aggregation sheets, preprint, 2012.
76. V.I. Yudovich, Non-stationary flow of an incompressible liquid, Zh. Vychisl. Mat. Mat. Fiz, 3 (1963), pp. 1032-1066.
77. Y. Zheng, Concentration-cancellation for the velocity fields in two dimensional incompressible fluid flows, Commun. Math. Phys., 135 (1991), pp. 581-594.