We study first order evolutive Mean Field Games where the Hamiltonian is non-coercive. This situation occurs, for instance, when some directions are "forbidden" to the generic player at some points. We establish the existence of a weak solution of the system via a vanishing viscosity method and, mainly, we prove that the evolution of the population's density is the push-forward of the initial density through the flow characterized almost everywhere by the optimal trajectories of the control problem underlying the Hamilton-Jacobi equation. As preliminary steps, we need that the optimal trajectories for the control problem are unique (at least for a.e. starting points) and that the optimal controls can be expressed in terms of the horizontal gradient of the value function.

中文翻译：

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非强制性一阶平均场博弈

我们研究一阶演化平均场博弈，其中哈密顿量是非强制性的。例如，当某些方向在某些点“禁止”通向一般玩家时，就会发生这种情况。我们通过粘度消失方法建立了系统弱解的存在性，并且主要证明了种群密度的演变是初始密度通过流动的推进，几乎处处都以最优轨迹为特征。 Hamilton-Jacobi 方程下的控制问题。作为初步步骤，我们需要控制问题的最优轨迹是唯一的（至少对于 ae 起始点），并且最优控制可以用价值函数的水平梯度来表示。