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Short‐time existence of solutions for mean‐field games with congestion
Author(s) -
Gomes Diogo A.,
Voskanyan Vardan K.
Publication year - 2015
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdv052
Subject(s) - uniqueness , hamilton–jacobi equation , mathematics , quadratic equation , field (mathematics) , fokker–planck equation , mathematical economics , mathematical analysis , partial differential equation , pure mathematics , geometry
We consider time‐dependent mean‐field games with congestion that are given by a Hamilton–Jacobi equation coupled with a Fokker–Planck equation. These models are motivated by crowd dynamics in which agents have difficulty moving in high‐density areas. The congestion effects make the Hamilton–Jacobi equation singular. The uniqueness of solutions for this problem is well understood; however, the existence of classical solutions was only known in very special cases, stationary problems with quadratic Hamiltonians and some time‐dependent explicit examples. Here, we demonstrate the short‐time existence of C ∞ solutions for sub‐quadratic Hamiltonians.