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Evolution Equations on Co-evolving Graphs: Long-Time Behaviour and the Graph-Continuity Equation.

José Antonio Carrillo1, Antonio Esposito2, László Mikolás1

  • 1Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG UK.

Journal of Nonlinear Science
|March 9, 2026
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Summary
This summary is machine-generated.

This study links evolution equations on co-evolving infinite graphs to nonlinear continuity equations. Researchers prove long-time convergence of solutions to uniform mass distribution using upwinding dynamics on graphs.

Keywords:
Co-evolving graphsEvolution on graphsLong-time behaviourNonlocal equations

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Area of Science:

  • Mathematics
  • Dynamical Systems
  • Graph Theory

Background:

  • Evolution equations on infinite graphs are crucial in modeling complex systems.
  • Understanding the behavior of these systems requires analyzing their dynamic evolution.
  • Nonlinear continuity equations provide a framework for describing mass distribution changes.

Purpose of the Study:

  • To establish a rigorous mathematical link between evolution equations on co-evolving infinite graphs and nonlinear continuity equations.
  • To analyze the behavior of weak solutions for graph-continuity equations.
  • To investigate the long-time convergence of solutions under specific dynamics.

Main Methods:

  • Establishing a connection between weak solutions and the flow map of associated characteristic equations.
  • Utilizing the push-forward of initial data through the flow map.
  • Applying upwinding dynamics with pointwise and monotonic velocity on graphs.

Main Results:

  • Weak solutions of graph-continuity equations are shown to be the push-forward of initial data.
  • A contraction in a suitable distance can be proven, despite limitations on flux.
  • Long-time convergence of solutions towards uniform mass distribution is demonstrated.

Conclusions:

  • The established link provides a powerful tool for analyzing evolution equations on dynamic graphs.
  • Upwinding dynamics offer a viable method for proving long-time convergence.
  • The findings contribute to the understanding of mass distribution in evolving graph-based systems.