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.
Highlights
This paper presents a survey of several results obtained by the authors concerning existence, uniqueness and numerical simulation of measure solutions for the one-dimensional aggregation equation in the attractive case
If ρ denotes the density of individuals, its dynamics is modelled by a nonlocal nonlinear conservation equation
Since we focus on scalar conservation laws, we can assume without loss of generality that the total mass of the system is scaled to 1
Summary
This paper presents a survey of several results obtained by the authors concerning existence, uniqueness and numerical simulation of measure solutions for the one-dimensional aggregation equation in the attractive case. This equation describes aggregation phenomena in a population of individuals interacting under a continuous potential W : R → R. In [9], global existence of weak measure solutions in the linear case, that is for W satisfying (1.2), in Rd for any dimension d ≥ 1 has been obtained using the gradient flow structure of this problem. Numerical illustrations showing different behaviours of solutions after blow up for different choices of the interaction potential are provided in subsection 5.4
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
