Existence of Solutions and Selection Problem for Quasi-stationary Contact Mean Field Games
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915388576497664 |
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| author | Hu, Xiaotian |
| author_facet | Hu, Xiaotian |
| contents | 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 $λ>0$, let $(u^λ,m^λ)$ be a solution of the system \begin{equation*} \left\{ \begin{aligned} &H(x,λu^λ,Du^λ)=F(x,m^λ(t))+c(m^λ(t)),\quad &&x\in M,\ \forall t\in[0,T],\\ &\partial_t m^λ-\text{div}\left(m^λ\dfrac{\partial H}{\partial p}(x,λu^λ,Du^λ)\right)=0,\quad &&(x,t)\in M\times(0,T],\\ &m(0)=m_0, \end{aligned}\right. \end{equation*} where $c(m^λ(t))$ is the Mañé critical value of the Hamiltonian $H(x,0,p)-F(x,m^λ(t))$. We investigate the selection problem for the limit of $(u^λ,m^λ)$ as $λ$ tends to 0. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_10112 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of Solutions and Selection Problem for Quasi-stationary Contact Mean Field Games Hu, Xiaotian Analysis of PDEs Dynamical Systems 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 $λ>0$, let $(u^λ,m^λ)$ be a solution of the system \begin{equation*} \left\{ \begin{aligned} &H(x,λu^λ,Du^λ)=F(x,m^λ(t))+c(m^λ(t)),\quad &&x\in M,\ \forall t\in[0,T],\\ &\partial_t m^λ-\text{div}\left(m^λ\dfrac{\partial H}{\partial p}(x,λu^λ,Du^λ)\right)=0,\quad &&(x,t)\in M\times(0,T],\\ &m(0)=m_0, \end{aligned}\right. \end{equation*} where $c(m^λ(t))$ is the Mañé critical value of the Hamiltonian $H(x,0,p)-F(x,m^λ(t))$. We investigate the selection problem for the limit of $(u^λ,m^λ)$ as $λ$ tends to 0. |
| title | Existence of Solutions and Selection Problem for Quasi-stationary Contact Mean Field Games |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2507.10112 |