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

François  James; Nicolas  Vauchelet; François  James; Nicolas  Vauchelet

doi:10.3934/nhm.2016.11.163

Networks and Heterogeneous Media

2016, Volume 11, Issue 1: 163-180. doi: 10.3934/nhm.2016.11.163

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

François James ^{1
,},
Nicolas Vauchelet ^{2
,}

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

Received: 01 April 2015 Revised: 01 September 2015
Primary: 35B40, 35D30, 35L60, 35Q92; Secondary: 49K20.

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.
- Aggregation equation,
- weak measure solutions,
- transport equation,
- blow up,
- finite volume scheme
Citation: François James, Nicolas Vauchelet. One-dimensional aggregation equation after blow up: Existence, uniqueness and numerical simulation[J]. Networks and Heterogeneous Media, 2016, 11(1): 163-180. doi: 10.3934/nhm.2016.11.163

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Abstract

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.

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