A Priori Bounds for Time-Dependent Models

We continue our study of the regularity of MFGs by considering the time-dependent problem (formula presented) acts as a perturbation of the heat equation and the main regularity tool is the Gagliardo–Nirenberg inequality. In the second instance, the Hopf–Cole transformation gives an explicit way to study (8.1). However, this transformation cannot be used to superquadratic problems. As a consequence, here, we use a technique that extends for superquadratic problems, 2, based on the nonlinear adjoint method. In the next chapter, we investigate two time-dependent problems with singularities—the logarithmic nonlinearity and the congestion problem—for which different methods are required.