A Priori Bounds for Time-Dependent Models

Abstract

We continue our study of the regularity of MFGs by considering the time-dependent problem

$$\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} -u_{t} + \frac{1} {\gamma } \left \vert Du\right \vert ^{\gamma } = \Delta u + m^{\alpha } \quad &\;\;\;\mbox{ in}\;\;\;\mathbb{T}^{d} \times [0,T], \\ m_{t} -\mathop{\mathrm{div}}\nolimits (\left \vert Du\right \vert ^{\gamma -1}m) = \Delta m\quad &\;\;\;\mbox{ in}\;\;\;\mathbb{T}^{d} \times [0,T], \end{array} \right. }$$

where 1 < γ ≤ 2 and α > 0. For γ < 2, we are in the subquadratic case; for γ = 2 the quadratic case. In the first instance, the non-linearity | Du | ^γ 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, γ > 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.