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Synchronising and nonsynchronising dynamics for a twospecies aggregation model
1.  Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire JacquesLouis Lions, F75005, Paris, France 
2.  CNRS, UMR 7598, Laboratoire JacquesLouis Lions, F75005, Paris, France 
3.  INRIAParisRocquencourt, EPC MAMBA, Domaine de Voluceau, BP105,78153 Le Chesnay Cedex, France 
4.  Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China 
5.  LAGA, UMR 7539, Institut Galilée, Université Paris 13, 99, avenue JeanBaptiste Clément, 93430 Villetaneuse, France 
This paper deals with analysis and numerical simulations of a onedimensional twospecies hyperbolic aggregation model. This model is formed by a system of transport equations with nonlocal velocities, which describes the aggregate dynamics of a twospecies population in interaction appearing for instance in bacterial chemotaxis. Blowup of classical solutions occurs in finite time. This raises the question to define measurevalued solutions for this system. To this aim, we use the duality method developed for transport equations with discontinuous velocity to prove the existence and uniqueness of measurevalued solutions. The proof relies on a stability result. In addition, this approach allows to study the hyperbolic limit of a kinetic chemotaxis model. Moreover, we propose a finite volume numerical scheme whose convergence towards measurevalued solutions is proved. It allows for numerical simulations capturing the behaviour after blow up. Finally, numerical simulations illustrate the complex dynamics of aggregates until the formation of a single aggregate: after blowup of classical solutions, aggregates of different species are synchronising or nonsynchronising when collide, that is move together or separately, depending on the parameters of the model and masses of species involved.
References:
[1] 
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, second ed. , Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. Google Scholar 
[2] 
A. L. Bertozzi and J. Brandman, Finitetime blowup of L^{∞}weak solutions of an aggregation equation, Commun. Math. Sci., 8 (2010), 4565. Google Scholar 
[3] 
F. Bouchut and F. James, Onedimensional transport equations with discontinuous coefficients, Nonlinear Anal., 32 (1998), 891933. Google Scholar 
[4] 
F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Equations, 24 (1999), 21732189. Google Scholar 
[5] 
M. CamposPinto, J. A. Carrillo, F. Charles and Y. P. Choi, Convergence of linearly transformed particle methods for the aggregation equation, submitted. Google Scholar 
[6] 
J. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepčev, Globalintime weak measure solutions and finitetime aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229271. Google Scholar 
[7] 
J. A. Carrillo, A. Chertock and Y. Huang, A finitevolume method for nonlinear nonlocal equations with a gradient flow structure, Communications in Computational Physics, 17 (2015), 233258. Google Scholar 
[8] 
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), 304338. Google Scholar 
[9] 
K. Craig and A. Bertozzi, A blob method for the aggregation equation, Math. Comp., 85 (2016), 16811717. Google Scholar 
[10] 
G. Crippa and M. LécureuxMercier, Existence and uniqueness of measure solutions for a system of continuity equations with nonlocal flow, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 523537. Google Scholar 
[11] 
M. Di Francesco and S. Fagioli, Measure solutions for nonlocal interaction PDEs with two species, Nonlinearity, 26 (2013), 27772808. Google Scholar 
[12] 
Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatiotemporal mechanisms, J. Math. Biol., 51 (2005), 595615. Google Scholar 
[13] 
C. Emako, C. Gayrard, A. Buguin, L. N. de Almeida and N. Vauchelet, Traveling pulses for a twospecies chemotaxis model, PLoS Comput. Biol., 12 (2016), e1004843. Google Scholar 
[14] 
C. Emako, L. Neves de Almeida and N. Vauchelet, Existence and diffusive limit of a twospecies kinetic model of chemotaxis, Kinetic and Related Models, 8 (2015), 359380. Google Scholar 
[15] 
L. Gosse and F. James, Numerical approximations of onedimensional linear conservation equations with discontinuous coefficients, Math. Comp., 69 (2000), 9871015. Google Scholar 
[16] 
D. Helbing, W. Yu and H. Rauhut, Selforganization and emergence in social systems: Modeling the coevolution of social environments and cooperative behavior, J. Math. Sociol., 35 (2011), 177208. Google Scholar 
[17] 
D. D Holm and V. Putkaradze, Aggregation of finitesize particles with variable mobility, Physical Review Letters, 95 (2005), 226106. Google Scholar 
[18] 
F. James and N. Vauchelet, Chemotaxis: From kinetic equations to aggregate dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101127. Google Scholar 
[19] 
F. James and N. Vauchelet, Numerical methods for onedimensional aggregation equations, SIAM Journal on Numerical Analysis, 53 (2015), 895916. Google Scholar 
[20] 
F. James and N. Vauchelet, Equivalence between duality and gradient flow solutions for onedimensional aggregation equations, Discrete Contin. Dyn. Syst., 36 (2016), 13551382. Google Scholar 
[21] 
T. Liu, M. L. K. Langston, D. Li, J. M. Pigga, C. Pichon, A. M. Todea and A. Müller, Selfrecognition among different polyprotic macroions during assembly processes in dilute solution, Science, 331 (2011), 15901592. Google Scholar 
[22] 
A. Mackey, T. Kolokolnikov and A. Bertozzi, Twospecies particle aggregation and stability of codimension one solutions, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 14111436. Google Scholar 
[23] 
N. Mittal, E. O. Budrene, M. P. Brenner and A. van Oudenaarden, Motility of escherichia coli cells in clusters formed by chemotactic aggregation, Proceedings of the National Academy of Sciences, 100 (2003), 1325913263. Google Scholar 
[24] 
H. Othmer, S. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263298. Google Scholar 
[25] 
F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal., 9 (2002), 533561. Google Scholar 
[26] 
S. T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. Ⅱ, Probability and its Applications (New York), SpringerVerlag, New York, 1998, Applications. Google Scholar 
[27] 
J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses, PLoS Comput. Biol., 6 (2010), e1000890, 12pp. Google Scholar 
[28] 
K. SznajdWeron and J. Sznajd, Opinion evolution in closed community, International Journal of Modern Physics C, 11 (2000), 11571165. Google Scholar 
[29] 
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. Google Scholar 
[30] 
C. Villani, Optimal Transport, Old and New, Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences], vol. 338, SpringerVerlag, Berlin, 2009. Google Scholar 
[31] 
J. H. Von Brecht, D. Uminsky, T. Kolokolnikov and A. L. Bertozzi, Predicting pattern formation in particle interactions, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1140002, 31pp. Google Scholar 
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 ed. , Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. Google Scholar 
[2] 
A. L. Bertozzi and J. Brandman, Finitetime blowup of L^{∞}weak solutions of an aggregation equation, Commun. Math. Sci., 8 (2010), 4565. Google Scholar 
[3] 
F. Bouchut and F. James, Onedimensional transport equations with discontinuous coefficients, Nonlinear Anal., 32 (1998), 891933. Google Scholar 
[4] 
F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Equations, 24 (1999), 21732189. Google Scholar 
[5] 
M. CamposPinto, J. A. Carrillo, F. Charles and Y. P. Choi, Convergence of linearly transformed particle methods for the aggregation equation, submitted. Google Scholar 
[6] 
J. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepčev, Globalintime weak measure solutions and finitetime aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229271. Google Scholar 
[7] 
J. A. Carrillo, A. Chertock and Y. Huang, A finitevolume method for nonlinear nonlocal equations with a gradient flow structure, Communications in Computational Physics, 17 (2015), 233258. Google Scholar 
[8] 
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), 304338. Google Scholar 
[9] 
K. Craig and A. Bertozzi, A blob method for the aggregation equation, Math. Comp., 85 (2016), 16811717. Google Scholar 
[10] 
G. Crippa and M. LécureuxMercier, Existence and uniqueness of measure solutions for a system of continuity equations with nonlocal flow, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 523537. Google Scholar 
[11] 
M. Di Francesco and S. Fagioli, Measure solutions for nonlocal interaction PDEs with two species, Nonlinearity, 26 (2013), 27772808. Google Scholar 
[12] 
Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatiotemporal mechanisms, J. Math. Biol., 51 (2005), 595615. Google Scholar 
[13] 
C. Emako, C. Gayrard, A. Buguin, L. N. de Almeida and N. Vauchelet, Traveling pulses for a twospecies chemotaxis model, PLoS Comput. Biol., 12 (2016), e1004843. Google Scholar 
[14] 
C. Emako, L. Neves de Almeida and N. Vauchelet, Existence and diffusive limit of a twospecies kinetic model of chemotaxis, Kinetic and Related Models, 8 (2015), 359380. Google Scholar 
[15] 
L. Gosse and F. James, Numerical approximations of onedimensional linear conservation equations with discontinuous coefficients, Math. Comp., 69 (2000), 9871015. Google Scholar 
[16] 
D. Helbing, W. Yu and H. Rauhut, Selforganization and emergence in social systems: Modeling the coevolution of social environments and cooperative behavior, J. Math. Sociol., 35 (2011), 177208. Google Scholar 
[17] 
D. D Holm and V. Putkaradze, Aggregation of finitesize particles with variable mobility, Physical Review Letters, 95 (2005), 226106. Google Scholar 
[18] 
F. James and N. Vauchelet, Chemotaxis: From kinetic equations to aggregate dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101127. Google Scholar 
[19] 
F. James and N. Vauchelet, Numerical methods for onedimensional aggregation equations, SIAM Journal on Numerical Analysis, 53 (2015), 895916. Google Scholar 
[20] 
F. James and N. Vauchelet, Equivalence between duality and gradient flow solutions for onedimensional aggregation equations, Discrete Contin. Dyn. Syst., 36 (2016), 13551382. Google Scholar 
[21] 
T. Liu, M. L. K. Langston, D. Li, J. M. Pigga, C. Pichon, A. M. Todea and A. Müller, Selfrecognition among different polyprotic macroions during assembly processes in dilute solution, Science, 331 (2011), 15901592. Google Scholar 
[22] 
A. Mackey, T. Kolokolnikov and A. Bertozzi, Twospecies particle aggregation and stability of codimension one solutions, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 14111436. Google Scholar 
[23] 
N. Mittal, E. O. Budrene, M. P. Brenner and A. van Oudenaarden, Motility of escherichia coli cells in clusters formed by chemotactic aggregation, Proceedings of the National Academy of Sciences, 100 (2003), 1325913263. Google Scholar 
[24] 
H. Othmer, S. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263298. Google Scholar 
[25] 
F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal., 9 (2002), 533561. Google Scholar 
[26] 
S. T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. Ⅱ, Probability and its Applications (New York), SpringerVerlag, New York, 1998, Applications. Google Scholar 
[27] 
J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses, PLoS Comput. Biol., 6 (2010), e1000890, 12pp. Google Scholar 
[28] 
K. SznajdWeron and J. Sznajd, Opinion evolution in closed community, International Journal of Modern Physics C, 11 (2000), 11571165. Google Scholar 
[29] 
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. Google Scholar 
[30] 
C. Villani, Optimal Transport, Old and New, Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences], vol. 338, SpringerVerlag, Berlin, 2009. Google Scholar 
[31] 
J. H. Von Brecht, D. Uminsky, T. Kolokolnikov and A. L. Bertozzi, Predicting pattern formation in particle interactions, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1140002, 31pp. Google Scholar 
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