One-dimensional, forward-forward mean-field games with congestion

Diogo Gomes*, Marc Sedjro

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


Here, we consider one-dimensional forward-forward mean-field games (MFGs) with congestion, which were introduced to approximate stationary MFGs. We use methods from the theory of conservation laws to examine the qualitative properties of these games. First, by computing Riemann invariants and corresponding invariant regions, we develop a method to prove lower bounds for the density. Next, by combining the lower bound with an entropy function, we prove the existence of global solutions for parabolic forwardforward MFGs. Finally, we construct traveling-wave solutions, which settles in a negative way the convergence problem for forward-forward MFGs. A similar technique gives the existence of time-periodic solutions for non-monotonic MFGs.

Original languageEnglish (US)
Pages (from-to)901-914
Number of pages14
JournalDiscrete and Continuous Dynamical Systems - Series S
Issue number5
StatePublished - Oct 2018

Bibliographical note

Funding Information:
91A15. Key words and phrases. Mean-field games, systems of conservation laws, convergence to equilibrium, Hamilton-Jacobi equations, transport equations, Fokker-Planck equations. D. Gomes was partially supported by KAUST baseline funds and KAUSTOSR-CRG2017-3452. M. Sedjro was supported by KAUST baseline and start-up funds. ∗ Corresponding author: Diogo Gomes.

Publisher Copyright:
© 2018 American Institute of Mathematical Sciences. All rights reserved.


  • Convergence to equilibrium
  • Fokker-Planck equations
  • Hamilton-Jacobi equations
  • Mean-field games
  • Systems of conservation laws
  • Transport equations

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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