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March  2016, 11(1): 163-180. doi: 10.3934/nhm.2016.11.163

## One-dimensional aggregation equation after blow up: Existence, uniqueness and numerical simulation

 1 Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Université d'Orléans & CNRS UMR 7349, Fédération Denis Poisson, Université d'Orléans & CNRS FR 2964, 45067 Orléans Cedex 2 2 Sorbonne Universites, UPMC Univ Paris 06, Laboratoire Jacques-Louis Lions UMR CNRS 7598, Inria, F-75005, Paris, France

Received  April 2015 Revised  September 2015 Published  January 2016

The nonlocal nonlinear aggregation equation in one space dimension is investigated. In the so-called attractive case smooth solutions blow up in finite time, so that weak measure solutions are introduced. The velocity involved in the equation becomes discontinuous, and a particular care has to be paid to its definition as well as the formulation of the corresponding flux. When this is done, the notion of duality solutions allows to obtain global in time existence and uniqueness for measure solutions. An upwind finite volume scheme is also analyzed, and the convergence towards the unique solution is proved. Numerical examples show the dynamics of the solutions after the blow up time.
Citation: François James, Nicolas Vauchelet. One-dimensional aggregation equation after blow up: Existence, uniqueness and numerical simulation. Networks and Heterogeneous Media, 2016, 11 (1) : 163-180. doi: 10.3934/nhm.2016.11.163
##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Space of Probability Measures, Lectures in Mathematics, Birkäuser, 2005. [2] D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media, RAIRO Model. Math. Anal. Numer., 31 (1997), 615-641. [3] A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equation with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710. doi: 10.1088/0951-7715/22/3/009. [4] S. Bianchini and M. Gloyer, An estimate on the flow generated by monotone operators, Comm. Partial Diff. Eq., 36 (2011), 777-796. doi: 10.1080/03605302.2010.534224. [5] 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. doi: 10.1016/j.jde.2005.07.025. [6] F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Analysis TMA, 32 (1998), 891-933. doi: 10.1016/S0362-546X(97)00536-1. [7] F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Eq., 24 (1999), 2173-2189. doi: 10.1080/03605309908821498. [8] J. A. Carrillo, A. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Comm. in Comp. Phys., 17 (2015), 233-258. doi: 10.4208/cicp.160214.010814a. [9] 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), 229-271. doi: 10.1215/00127094-2010-211. [10] J. A. Carrillo, F. James, F. Lagoutière and N. Vauchelet, The Filippov characteristic flow for the aggregation equation with mildly singular potentials, J. Differential Equations, 260 (2016), 304-338. doi: 10.1016/j.jde.2015.08.048. [11] R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34p. doi: 10.1142/S0218202511500230. [12] K. Craig and A. L. Bertozzi, A blob method for the aggregation equation, Math. Comp., (2015), 1-37. doi: 10.1090/mcom3033. [13] Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms, J. Math. Biol., 51 (2005), 595-615. doi: 10.1007/s00285-005-0334-6. [14] F. Filbet, Ph. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207. doi: 10.1007/s00285-004-0286-2. [15] E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences 118, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0713-9. [16] L. Gosse and F. James, Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients, Math. Comput., 69 (2000), 987-1015. doi: 10.1090/S0025-5718-00-01185-6. [17] A. Harten, On a class of high resolution total-variation-stable finite difference schemes, SIAM Jour. of Numer. Anal., 21 (1984), 1-23. doi: 10.1137/0721001. [18] F. James and N. Vauchelet, A remark on duality solutions for some weakly nonlinear scalar conservation laws, C. R. Acad. Sci. Paris, Sér. I, 349 (2011), 657-661. doi: 10.1016/j.crma.2011.05.004. [19] F. James and N. Vauchelet, Chemotaxis: from kinetic equations to aggregation dynamics, Nonlinear Diff. Eq. and Appl. (NoDEA), 20 (2013), 101-127. doi: 10.1007/s00030-012-0155-4. [20] F. James and N. Vauchelet, Equivalence between duality and gradient flow solutions for one-dimensional aggregation equations, Disc. Cont. Dyn. Syst., 36 (2016), 1355-1382. [21] F. James and N. Vauchelet, Numerical method for one-dimensional aggregation equations, SIAM J. Numer. Anal., 53 (2015), 895-916. doi: 10.1137/140959997. [22] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359. [23] A.-Y. Le Roux, A numerical conception of entropy for quasi-linear equations, Math. of Comp., 31 (1977), 848-872. doi: 10.1090/S0025-5718-1977-0478651-3. [24] H. Li and G. Toscani, Long time asymptotics of kinetic models of granular flows, Arch. Rat. Mech. Anal., 172 (2004), 407-428. doi: 10.1007/s00205-004-0307-8. [25] B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic Crowd Motion Model of the gradient-flow type, Math. Models and Methods in Applied Sci., 20 (2010), 1787-1821. doi: 10.1142/S0218202510004799. [26] J. Nieto, F. Poupaud and J. Soler, High field limit for Vlasov-Poisson-Fokker-Planck equations, Arch. Rational Mech. Anal., 158 (2001), 29-59. doi: 10.1007/s002050100139. [27] A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives, Springer, Berlin, 2001. doi: 10.1007/978-1-4757-4978-6. [28] F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal., 9 (2002), 533-561. doi: 10.4310/MAA.2002.v9.n4.a4. [29] F. Poupaud and M. Rascle, Measure solutions to the linear multidimensional transport equation with discontinuous coefficients, Comm. Partial Diff. Equ., 22 (1997), 337-358. doi: 10.1080/03605309708821265. [30] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58, Amer. Math. Soc, Providence, 2003. doi: 10.1007/b12016.

show all references

##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Space of Probability Measures, Lectures in Mathematics, Birkäuser, 2005. [2] D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media, RAIRO Model. Math. Anal. Numer., 31 (1997), 615-641. [3] A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equation with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710. doi: 10.1088/0951-7715/22/3/009. [4] S. Bianchini and M. Gloyer, An estimate on the flow generated by monotone operators, Comm. Partial Diff. Eq., 36 (2011), 777-796. doi: 10.1080/03605302.2010.534224. [5] 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. doi: 10.1016/j.jde.2005.07.025. [6] F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Analysis TMA, 32 (1998), 891-933. doi: 10.1016/S0362-546X(97)00536-1. [7] F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Eq., 24 (1999), 2173-2189. doi: 10.1080/03605309908821498. [8] J. A. Carrillo, A. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Comm. in Comp. Phys., 17 (2015), 233-258. doi: 10.4208/cicp.160214.010814a. [9] 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), 229-271. doi: 10.1215/00127094-2010-211. [10] J. A. Carrillo, F. James, F. Lagoutière and N. Vauchelet, The Filippov characteristic flow for the aggregation equation with mildly singular potentials, J. Differential Equations, 260 (2016), 304-338. doi: 10.1016/j.jde.2015.08.048. [11] R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34p. doi: 10.1142/S0218202511500230. [12] K. Craig and A. L. Bertozzi, A blob method for the aggregation equation, Math. Comp., (2015), 1-37. doi: 10.1090/mcom3033. [13] Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms, J. Math. Biol., 51 (2005), 595-615. doi: 10.1007/s00285-005-0334-6. [14] F. Filbet, Ph. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207. doi: 10.1007/s00285-004-0286-2. [15] E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences 118, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0713-9. [16] L. Gosse and F. James, Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients, Math. Comput., 69 (2000), 987-1015. doi: 10.1090/S0025-5718-00-01185-6. [17] A. Harten, On a class of high resolution total-variation-stable finite difference schemes, SIAM Jour. of Numer. Anal., 21 (1984), 1-23. doi: 10.1137/0721001. [18] F. James and N. Vauchelet, A remark on duality solutions for some weakly nonlinear scalar conservation laws, C. R. Acad. Sci. Paris, Sér. I, 349 (2011), 657-661. doi: 10.1016/j.crma.2011.05.004. [19] F. James and N. Vauchelet, Chemotaxis: from kinetic equations to aggregation dynamics, Nonlinear Diff. Eq. and Appl. (NoDEA), 20 (2013), 101-127. doi: 10.1007/s00030-012-0155-4. [20] F. James and N. Vauchelet, Equivalence between duality and gradient flow solutions for one-dimensional aggregation equations, Disc. Cont. Dyn. Syst., 36 (2016), 1355-1382. [21] F. James and N. Vauchelet, Numerical method for one-dimensional aggregation equations, SIAM J. Numer. Anal., 53 (2015), 895-916. doi: 10.1137/140959997. [22] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359. [23] A.-Y. Le Roux, A numerical conception of entropy for quasi-linear equations, Math. of Comp., 31 (1977), 848-872. doi: 10.1090/S0025-5718-1977-0478651-3. [24] H. Li and G. Toscani, Long time asymptotics of kinetic models of granular flows, Arch. Rat. Mech. Anal., 172 (2004), 407-428. doi: 10.1007/s00205-004-0307-8. [25] B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic Crowd Motion Model of the gradient-flow type, Math. Models and Methods in Applied Sci., 20 (2010), 1787-1821. doi: 10.1142/S0218202510004799. [26] J. Nieto, F. Poupaud and J. Soler, High field limit for Vlasov-Poisson-Fokker-Planck equations, Arch. Rational Mech. Anal., 158 (2001), 29-59. doi: 10.1007/s002050100139. [27] A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives, Springer, Berlin, 2001. doi: 10.1007/978-1-4757-4978-6. [28] F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal., 9 (2002), 533-561. doi: 10.4310/MAA.2002.v9.n4.a4. [29] F. Poupaud and M. Rascle, Measure solutions to the linear multidimensional transport equation with discontinuous coefficients, Comm. Partial Diff. Equ., 22 (1997), 337-358. doi: 10.1080/03605309708821265. [30] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58, Amer. Math. Soc, Providence, 2003. doi: 10.1007/b12016.
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