拥挤平均场对策的弱解
Weak solutions for mean field games with congestion
摘要：
研究了平均场对策论中一类耦合的偏微分方程组解的短时存在唯一性。它的一般形式为$$ \left\{ \begin{array}{c} -\partial_t u - Δu + H(t,x,m,\nabla u) = f(t,x,m) \\ \partial_t m - Δm - \mathrm{div} \left(m\nabla_p H(t,x,m,\nabla u)\right) = 0 \end{array}\right. $$加上初末和边界条件。该问题的新颖性在于，对于某些$α\geq 0$和$r > 1$，哈密顿量$H(t,x,m,p)$可以采取诸如$m^{-α}|p|^r$的形式。我们的主要结果是只要$r$不是太大，小时间$T$的弱解的存在性，以及在附加约束下的唯一性.证明的主要内容是对福克-普朗克方程解的先验估计。我们还简要地考虑了平均场对策相关的最优控制问题的解的存在性和唯一性。
英文摘要：
We study the short-time existence and uniqueness of solutions to a coupled system of partial differential equations arising in mean field game theory. It has the generic form $$ \left\{ \begin{array}{c} -\partial_t u - Δu + H(t,x,m,\nabla u) = f(t,x,m) \\ \partial_t m - Δm - \mathrm{div} \left(m\nabla_p H(t,x,m,\nabla u)\right) = 0 \end{array}\right. $$ plus initial-final and boundary conditions. The novelty of the problem is that the Hamiltonian $H(t,x,m,p)$ may take such forms as $m^{-α}|p|^r$ for some $α\geq 0$ and $r > 1$. Our main result is the existence of weak solutions for small times $T$ so long as $r$ is not too large, and uniqueness under additional constraints. The main ingredient in the proof is an a priori estimate on solutions to the Fokker-Planck equation. We also briefly consider existence and uniqueness of solutions to an optimal control problem related to mean field games.
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