Existence and Structure for First-Order Time-Dependent Mean-Field Games with Local Couplings

We develop a Banach-space framework for first-order time-dependent mean-field games with local couplings, using monotone operator theory and low-order $p$-Laplacian regularization to avoid high-order elliptic smoothing. Under monotonicity and power-growth assumptions, together with either a Lagrangian lower bound or strict positivity of the initial density, we prove existence of weak variational-inequality solutions by Minty's method. The constructed solutions satisfy uniform $L^\beta$ estimates on the density, $L^\alpha$ estimates on the spatial gradient of the value function, and space-time shift estimates sufficient to identify the limiting PDE system. We prove that any variational-inequality solution satisfying these bounds, regardless of how it is obtained, is a MFG solution satisfying the Hamilton--Jacobi and transport equations in the $BV$ sense. This separates the construction of VI-solutions from the verification of the PDE system, a feature not directly available in the existing stationary Banach-space framework. Finally, for each fixed density $m$, we establish a maximal value function among Hamilton--Jacobi subsolutions; every MFG value function coincides with this maximal representative on $\{m>0\}$ and initially on $\{m_0>0\}$. Under semi-strict monotonicity, the density $m$ itself is unique. Our results apply to non-separable Hamiltonians with power growth and impose no dimension restrictions.