• Previous Article
    On a hyperbolic Keller-Segel system with degenerate nonlinear fractional diffusion
  • NHM Home
  • This Issue
  • Next Article
    Entropy stability and well-balancedness of space-time DG for the shallow water equations with bottom topography
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 Universites, 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 & 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, (2005). Google Scholar

[2]

D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media,, RAIRO Model. Math. Anal. Numer., 31 (1997), 615. Google Scholar

[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. doi: 10.1088/0951-7715/22/3/009. Google Scholar

[4]

S. Bianchini and M. Gloyer, An estimate on the flow generated by monotone operators,, Comm. Partial Diff. Eq., 36 (2011), 777. doi: 10.1080/03605302.2010.534224. Google Scholar

[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. doi: 10.1016/j.jde.2005.07.025. Google Scholar

[6]

F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients,, Nonlinear Analysis TMA, 32 (1998), 891. doi: 10.1016/S0362-546X(97)00536-1. Google Scholar

[7]

F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness,, Comm. Partial Differential Eq., 24 (1999), 2173. doi: 10.1080/03605309908821498. Google Scholar

[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. doi: 10.4208/cicp.160214.010814a. Google Scholar

[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. doi: 10.1215/00127094-2010-211. Google Scholar

[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. doi: 10.1016/j.jde.2015.08.048. Google Scholar

[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). doi: 10.1142/S0218202511500230. Google Scholar

[12]

K. Craig and A. L. Bertozzi, A blob method for the aggregation equation,, Math. Comp., (2015), 1. doi: 10.1090/mcom3033. Google Scholar

[13]

Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms,, J. Math. Biol., 51 (2005), 595. doi: 10.1007/s00285-005-0334-6. Google Scholar

[14]

F. Filbet, Ph. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, J. Math. Biol., 50 (2005), 189. doi: 10.1007/s00285-004-0286-2. Google Scholar

[15]

E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws,, Applied Mathematical Sciences 118, 118 (1996). doi: 10.1007/978-1-4612-0713-9. Google Scholar

[16]

L. Gosse and F. James, Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients,, Math. Comput., 69 (2000), 987. doi: 10.1090/S0025-5718-00-01185-6. Google Scholar

[17]

A. Harten, On a class of high resolution total-variation-stable finite difference schemes,, SIAM Jour. of Numer. Anal., 21 (1984), 1. doi: 10.1137/0721001. Google Scholar

[18]

F. James and N. Vauchelet, A remark on duality solutions for some weakly nonlinear scalar conservation laws,, C. R. Acad. Sci. Paris, 349 (2011), 657. doi: 10.1016/j.crma.2011.05.004. Google Scholar

[19]

F. James and N. Vauchelet, Chemotaxis: from kinetic equations to aggregation dynamics,, Nonlinear Diff. Eq. and Appl. (NoDEA), 20 (2013), 101. doi: 10.1007/s00030-012-0155-4. Google Scholar

[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. Google Scholar

[21]

F. James and N. Vauchelet, Numerical method for one-dimensional aggregation equations,, SIAM J. Numer. Anal., 53 (2015), 895. doi: 10.1137/140959997. Google Scholar

[22]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1. doi: 10.1137/S0036141096303359. Google Scholar

[23]

A.-Y. Le Roux, A numerical conception of entropy for quasi-linear equations,, Math. of Comp., 31 (1977), 848. doi: 10.1090/S0025-5718-1977-0478651-3. Google Scholar

[24]

H. Li and G. Toscani, Long time asymptotics of kinetic models of granular flows,, Arch. Rat. Mech. Anal., 172 (2004), 407. doi: 10.1007/s00205-004-0307-8. Google Scholar

[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. doi: 10.1142/S0218202510004799. Google Scholar

[26]

J. Nieto, F. Poupaud and J. Soler, High field limit for Vlasov-Poisson-Fokker-Planck equations,, Arch. Rational Mech. Anal., 158 (2001), 29. doi: 10.1007/s002050100139. Google Scholar

[27]

A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives,, Springer, (2001). doi: 10.1007/978-1-4757-4978-6. Google Scholar

[28]

F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation,, Methods Appl. Anal., 9 (2002), 533. doi: 10.4310/MAA.2002.v9.n4.a4. Google Scholar

[29]

F. Poupaud and M. Rascle, Measure solutions to the linear multidimensional transport equation with discontinuous coefficients,, Comm. Partial Diff. Equ., 22 (1997), 337. doi: 10.1080/03605309708821265. Google Scholar

[30]

C. Villani, Topics in Optimal Transportation,, Graduate Studies in Mathematics 58, 58 (2003). doi: 10.1007/b12016. Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Space of Probability Measures,, Lectures in Mathematics, (2005). Google Scholar

[2]

D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media,, RAIRO Model. Math. Anal. Numer., 31 (1997), 615. Google Scholar

[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. doi: 10.1088/0951-7715/22/3/009. Google Scholar

[4]

S. Bianchini and M. Gloyer, An estimate on the flow generated by monotone operators,, Comm. Partial Diff. Eq., 36 (2011), 777. doi: 10.1080/03605302.2010.534224. Google Scholar

[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. doi: 10.1016/j.jde.2005.07.025. Google Scholar

[6]

F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients,, Nonlinear Analysis TMA, 32 (1998), 891. doi: 10.1016/S0362-546X(97)00536-1. Google Scholar

[7]

F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness,, Comm. Partial Differential Eq., 24 (1999), 2173. doi: 10.1080/03605309908821498. Google Scholar

[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. doi: 10.4208/cicp.160214.010814a. Google Scholar

[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. doi: 10.1215/00127094-2010-211. Google Scholar

[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. doi: 10.1016/j.jde.2015.08.048. Google Scholar

[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). doi: 10.1142/S0218202511500230. Google Scholar

[12]

K. Craig and A. L. Bertozzi, A blob method for the aggregation equation,, Math. Comp., (2015), 1. doi: 10.1090/mcom3033. Google Scholar

[13]

Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms,, J. Math. Biol., 51 (2005), 595. doi: 10.1007/s00285-005-0334-6. Google Scholar

[14]

F. Filbet, Ph. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, J. Math. Biol., 50 (2005), 189. doi: 10.1007/s00285-004-0286-2. Google Scholar

[15]

E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws,, Applied Mathematical Sciences 118, 118 (1996). doi: 10.1007/978-1-4612-0713-9. Google Scholar

[16]

L. Gosse and F. James, Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients,, Math. Comput., 69 (2000), 987. doi: 10.1090/S0025-5718-00-01185-6. Google Scholar

[17]

A. Harten, On a class of high resolution total-variation-stable finite difference schemes,, SIAM Jour. of Numer. Anal., 21 (1984), 1. doi: 10.1137/0721001. Google Scholar

[18]

F. James and N. Vauchelet, A remark on duality solutions for some weakly nonlinear scalar conservation laws,, C. R. Acad. Sci. Paris, 349 (2011), 657. doi: 10.1016/j.crma.2011.05.004. Google Scholar

[19]

F. James and N. Vauchelet, Chemotaxis: from kinetic equations to aggregation dynamics,, Nonlinear Diff. Eq. and Appl. (NoDEA), 20 (2013), 101. doi: 10.1007/s00030-012-0155-4. Google Scholar

[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. Google Scholar

[21]

F. James and N. Vauchelet, Numerical method for one-dimensional aggregation equations,, SIAM J. Numer. Anal., 53 (2015), 895. doi: 10.1137/140959997. Google Scholar

[22]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1. doi: 10.1137/S0036141096303359. Google Scholar

[23]

A.-Y. Le Roux, A numerical conception of entropy for quasi-linear equations,, Math. of Comp., 31 (1977), 848. doi: 10.1090/S0025-5718-1977-0478651-3. Google Scholar

[24]

H. Li and G. Toscani, Long time asymptotics of kinetic models of granular flows,, Arch. Rat. Mech. Anal., 172 (2004), 407. doi: 10.1007/s00205-004-0307-8. Google Scholar

[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. doi: 10.1142/S0218202510004799. Google Scholar

[26]

J. Nieto, F. Poupaud and J. Soler, High field limit for Vlasov-Poisson-Fokker-Planck equations,, Arch. Rational Mech. Anal., 158 (2001), 29. doi: 10.1007/s002050100139. Google Scholar

[27]

A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives,, Springer, (2001). doi: 10.1007/978-1-4757-4978-6. Google Scholar

[28]

F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation,, Methods Appl. Anal., 9 (2002), 533. doi: 10.4310/MAA.2002.v9.n4.a4. Google Scholar

[29]

F. Poupaud and M. Rascle, Measure solutions to the linear multidimensional transport equation with discontinuous coefficients,, Comm. Partial Diff. Equ., 22 (1997), 337. doi: 10.1080/03605309708821265. Google Scholar

[30]

C. Villani, Topics in Optimal Transportation,, Graduate Studies in Mathematics 58, 58 (2003). doi: 10.1007/b12016. Google Scholar

[1]

Yong Zhou, Zhengguang Guo. Blow up and propagation speed of solutions to the DGH equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 657-670. doi: 10.3934/dcdsb.2009.12.657

[2]

Piotr Gwiazda, Sander C. Hille, Kamila Łyczek, Agnieszka Świerczewska-Gwiazda. Differentiability in perturbation parameter of measure solutions to perturbed transport equation. Kinetic & Related Models, 2019, 12 (5) : 1093-1108. doi: 10.3934/krm.2019041

[3]

Mohamed Alahyane, Abdelilah Hakim, Amine Laghrib, Said Raghay. Fluid image registration using a finite volume scheme of the incompressible Navier Stokes equation. Inverse Problems & Imaging, 2018, 12 (5) : 1055-1081. doi: 10.3934/ipi.2018044

[4]

Rajesh Kumar, Jitendra Kumar, Gerald Warnecke. Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations. Kinetic & Related Models, 2014, 7 (4) : 713-737. doi: 10.3934/krm.2014.7.713

[5]

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

[6]

Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089

[7]

István Győri, Yukihiko Nakata, Gergely Röst. Unbounded and blow-up solutions for a delay logistic equation with positive feedback. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2845-2854. doi: 10.3934/cpaa.2018134

[8]

Frédéric Abergel, Jean-Michel Rakotoson. Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1809-1818. doi: 10.3934/dcds.2013.33.1809

[9]

Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881

[10]

Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065

[11]

Futoshi Takahashi. Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data. Conference Publications, 2013, 2013 (special) : 729-736. doi: 10.3934/proc.2013.2013.729

[12]

Andrea L. Bertozzi, Dejan Slepcev. Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1617-1637. doi: 10.3934/cpaa.2010.9.1617

[13]

Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1

[14]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843

[15]

Yohan Penel. An explicit stable numerical scheme for the $1D$ transport equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 641-656. doi: 10.3934/dcdss.2012.5.641

[16]

Akmel Dé Godefroy. Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 117-137. doi: 10.3934/dcds.2015.35.117

[17]

Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085

[18]

Min Li, Zhaoyang Yin. Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280

[19]

Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831

[20]

Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903

2018 Impact Factor: 0.871

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]