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

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.