## Duke Mathematical Journal

### Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations

#### Abstract

In this paper we provide a well-posedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractive/repulsive pairwise interaction potentials. The main phenomenon of interest is that, even with smooth initial data, the solutions can concentrate mass in finite time. We develop an existence theory that enables one to go beyond the blow-up time in classical norms and allows for solutions to form atomic parts of the measure in finite time. The weak measure solutions are shown to be unique and exist globally in time. Moreover, in the case of sufficiently attractive potentials, we show the finite-time total collapse of the solution onto a single point for compactly supported initial measures. Our approach is based on the theory of gradient flows in the space of probability measures endowed with the Wasserstein metric. In addition to classical tools, we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations.

#### Article information

Source
Duke Math. J. Volume 156, Number 2 (2011), 229-271.

Dates
First available in Project Euclid: 2 February 2011

https://projecteuclid.org/euclid.dmj/1296662020

Digital Object Identifier
doi:10.1215/00127094-2010-211

Mathematical Reviews number (MathSciNet)
MR2769217

Zentralblatt MATH identifier
1215.35045

#### Citation

Carrillo, J. A.; DiFrancesco, M.; Figalli, A.; Laurent, T.; Slepčev, D. Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Duke Math. J. 156 (2011), no. 2, 229--271. doi:10.1215/00127094-2010-211. https://projecteuclid.org/euclid.dmj/1296662020

#### References

• M. Agueh, Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory, Adv. Differential Equations 10 (2005), 309–360.
• L. Ambrosio, N. Gigli, and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures Math. ETH Zürich, Birkhäuser, Basel, 2005.
• L. Ambrosio and G. Savaré, Gradient Flows of Probability Measures, Handb. Differ. Equ. 3, Elsevier/North-Holland, Amsterdam, 2006.
• D. Benedetto, E. Caglioti, and M. Pulvirenti, A kinetic equation for granular media, RAIRO Modél. Math. Anal. Numér. 31 (1997), 615–641.
• A. L. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation, Commun. Math. Sci. 8 (2010), 45–65.
• A. L. Bertozzi, J. A. Carrillo, and T. Laurent, Blowup in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity 22 (2009), 683–710.
• A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in $\R^n$, Comm. Math. Phys. 274 (2007), 717–735.
• A. L. Bertozzi, T. Laurent, and J. Rosado, $L^p$ Theory for the multidimensional aggregation equation, to appear in Comm. Pure Appl. Math.
• P. Biler, G. Karch, and P. Laurençot, Blowup of solutions to a diffusive aggregation model, Nonlinearity 22 (2009), 1559–1568.
• A. Blanchet, V. Calvez, and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model, SIAM J. Numer. Anal. 46 (2008), 691–721.
• A. Blanchet, J. A. Carrillo, and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\R^2$, Comm. Pure Appl. Math. 61 (2008), 1449–1481.
• 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 (2006), no. 44. MR2226917
• 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), 341–380.
• S. Boi, V. Capasso, and D. Morale, “Modeling the aggregative behavior of ants of the species Polyergus rufescens” in Spatial Heterogeneity in Ecological Models (Alcalá de Henares, Spain, 1998), Nonlinear Anal. Real World Appl. 1 (2000), 163–176.
• M. Burger, V. Capasso, and D. Morale, On an aggregation model with long and short range interactions, Nonlinear Anal. Real World Appl. 8 (2007), 939–958.
• M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Netw. Heterog. Media 3 (2008), 749–785.
• J. A. Carrillo, M. R. D'Orsogna, and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models 2 (2009), 363–378.
• 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), 971–1018.
• —, Contractions in the $2$-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal. 179 (2006), 217–263.
• 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, Eur. Math. Soc., Zürich, 2010, 3–16.
• Y.-L. Chuang, Y. R. Huang, M. R. D'Orsogna, and A. L. Bertozzi, “Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials” in 2007 IEEE International Conference on Robotics and Automation, IEEE, Piscataway, N.J., 2007, 2292–2299.
• R. Dobrushin, Vlasov equations, Funktsional. Anal. i Prilozhen. 13 (1979), 48–58; English translation in Functional Anal. Appl. 13 (1979), 115–123.
• F. Golse, The Mean-Field Limit for the Dynamics of Large Particle Systems, Journées “Équations aux Dérivées Partielles,” exp. no. IX, Univ. Nantes, Nantes, 2003.
• Y. Huang and A. L. Bertozzi, Self-similar blow-up solutions to an aggregation equation in $\real^n$, SIAM. J. Appl. Math. 70 (2010), 2582–2603.
• R. Jordan, D. Kinderlehrer, and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal. 29 (1998), 1–17.
• E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399–415.
• T. Laurent, Local and global existence for an aggregation equation, Comm. Partial Differential Equations 32 (2007), 1941–1964.
• D. Li and J. Rodrigo, Finite-time singularities of an aggregation equation in $\R^n$ with fractional dissipation, Comm. Math. Phys. 287 (2009), 687–703.
• D. Li and J. Rodrigo, Refined blowup criteria and nonsymmetric blowup of an aggregation equation, Adv. Math. 220 (2009), 1717–1738.
• D. Li and X. Zhang, On a nonlocal aggregation model with nonlinear diffusion, Discrete Contin. Dyn. Syst. 27 (2010), 301–323.
• H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Arch. Ration. Mech. Anal. 172 (2004), 407–428.
• R. J. Mccann, A convexity principle for interacting gases, Adv. Math. 128 (1997), 153–179.
• A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Bio. 38 (1999), 534–570.
• D. Morale, V. Capasso, and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol. 50 (2005), 49–66.
• H. Neunzert, The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles, Trans. Fluid Dynamics 18 (1977), 663–678.
• A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives, Interdiscip. Appl. Math. 14, Springer, Berlin, 2002.
• F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations 26 (2001), 101–174.
• C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys. 15 (1953), 311–338.
• F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal. 9 (2002), 533–561.
• H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. Modern Phys. 52 (1980), 569–615.
• C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math. 65 (2004), 152–174.
• C. M. Topaz, A. L. Bertozzi, and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol. 68 (2006), 1601–1623.
• G. Toscani, One-dimensional kinetic models of granular flows, M2AN Math. Model. Nummer. Anal. 34 (2000), 1277–1291.
• C. Villani, Topics in Optimal Transportation, Grad. Stud. Math., Amer. Math. Soc., Providence, 2003.
• —, Optimal Transport, Old and New, Grundlehren Math. Wiss. 338, Springer, Berlin, 2009.