Here, we establish the existence of weak solutions to a wide class of time-dependent monotone mean-field games (MFGs). These MFGs are given as a system of degenerate parabolic equations with initial and terminal conditions. To construct these solutions, we consider a high-order elliptic regularization in space–time. Then, applying Schaefer’s fixed-point theorem, we obtain the existence and uniqueness for this regularized problem. Using Minty’s method, we prove the existence of a weak solution to the original MFG. Finally, the paper ends with a discussion on congestion problems and density constrained MFGs.

中文翻译：

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时间相关平均场博弈弱解的存在

在这里，我们建立了一类广泛的依赖于时间的单调平均场博弈（MFG）的弱解的存在。这些 MFG 作为具有初始和终端条件的退化抛物线方程系统给出。为了构建这些解决方案，我们考虑了时空的高阶椭圆正则化。然后，应用谢弗不动点定理，我们得到这个正则化问题的存在唯一性。使用 Minty 的方法，我们证明了原始 MFG 的弱解的存在。最后，本文以对拥塞问题和密度约束 MFG 的讨论结束。