Existence of Solutions and Selection Problem for Quasi-stationary Contact Mean Field Games

TL;DR

This paper investigates the existence of solutions for a class of first order mean field games systems involving contact Hamilton-Jacobi equations and studies the selection problem for the limit of solutions as a parameter tends to zero.

Contribution

It establishes the existence of solutions for contact mean field games and analyzes the limiting behavior of solutions as a parameter approaches zero.

Findings

Proved existence of solutions for the mean field games system.

Analyzed the limit behavior of solutions as the parameter tends to zero.

Identified the selection mechanism for the limit solutions.

Abstract

First, we study the existence of solutions for a class of first order mean field games systems \begin{equation*} \left\{\begin{aligned} &H(x,u,Du)=F(x,m(t)),\quad &&x\in M,\ \forall\ t\in[0,T],\\ &\partial_t m-\text{div}\left(m\dfrac{\partial H}{\partial p}(x,u,Du)\right)=0,\quad &&(x,t)\in M\times(0,T],\\ &m(0)=m_0, \end{aligned}\right. \end{equation*} where the system comprises a stationary Hamilton-Jacobi equation in the contact case and an evolutionary continuity equation. Then, for any fixed $\lambda>0$, let $(u^\lambda,m^\lambda)$ be a solution of the system \begin{equation*} \left\{ \begin{aligned} &H(x,\lambda u^\lambda,Du^\lambda)=F(x,m^\lambda(t))+c(m^\lambda(t)),\quad &&x\in M,\ \forall t\in[0,T],\\ &\partial_t m^\lambda-\text{div}\left(m^\lambda\dfrac{\partial H}{\partial p}(x,\lambda u^\lambda,Du^\lambda)\right)=0,\quad &&(x,t)\in M\times(0,T],\\ &m(0)=m_0, \end{aligned}\right. \end{equation*} where $c(m^\lambda(t))$ is the Ma\~n\'e critical value of the Hamiltonian $H(x,0,p)-F(x,m^\lambda(t))$. We investigate the selection problem for the limit of $(u^\lambda,m^\lambda)$ as $\lambda$ tends to 0.