Short-Time Existence for a General Backward–Forward Parabolic System Arising from Mean-Field Games

Abstract

We study the local in time existence of a regular solution of a nonlinear parabolic backward–forward system arising from the theory of mean-field games (briefly MFG). The proof is based on a contraction argument in a suitable space that takes account of the peculiar structure of the system, which involves also a coupling at the final horizon. We apply the result to obtain existence to very general MFG models, including also congestion problems.

Appendix

In this final appendix, we prove Proposition 2.6 of Sect. 2.

Proof of Proposition 2.6

We write the proof for the existence of a solution in the class $C^{2+\alpha , 1+\alpha /2}(Q_T)$ and for the estimate (2.8). In a similar way one obtains the existence in $W^{2,1}_q(Q_T)$ and the proof of (2.9).

Recall that the problem on $\mathbb {T}^N\times [0,T]$ is equivalent to the same problem with 1-periodic data in the x-variable in ${\mathbb {R}}^N\times [0,T]$, namely with all the data satisfying $w(x+z,t)=w(x,t)$ for all $z\in {{I\!\!Z}}^N$. As far as the existence of a smooth solution of problem (2.7) is concerned, it is sufficient to apply Theorem 5.1 p.320 of [20]. Since the solution of such a Cauchy problem is unique, it must be periodic in the x-variable. Now we prove estimate (2.8). Let $R_1^N:=[-1,1]^N$ and $R_2^N := [-2,2]^N$. Clearly

$$\begin{aligned}{}[0,1]^N \subset R_1^N\subset R_2^N \subset {\mathbb {R}}^N \end{aligned}$$

(A.1)

and $dist(R_1^N, C(R_2^N))=1$.

We take advantage of local parabolic estimates, which allow us to get an a priori estimate regardless of the lateral boundary conditions which are unknown for us.

In particular, using the local estimate (10.5) p. 352 of [20] with $\Omega '=R_1^N$ and $\Omega '':={R_2^N}$, (note that in our case $S''$ is empty) we have

$$\begin{aligned} |u|^{(\alpha +2)}_{R_1^N\times [0,T^*]}\le C_1\left( |f|^{(\alpha )}_{R_2^N\times [0,T^*]}+ |u_0|^{(\alpha +2)}_{R_2^N}\right) + C_2 |u|_{R_2^N\times [0,T^*]}, \end{aligned}$$

(A.2)

where $T^*<T$, $C_1$ and $C_2$ depend on N, T, and the modulus of Hölder continuity of the coefficients of the operator. It is now crucial to observe that Hölder norms on $R_1^N\times [0,T^*]$, $R_2^N\times [0,T^*]$ and $\mathbb T^N\times [0,T^*]$ coincide by periodicity of u, f, $u_0$ in the x-variable and the inclusions (A.1). Hence,

$$\begin{aligned} |u|^{(\alpha +2)}_{\mathbb T^N\times [0,T^*]}\le C_1\left( |f|^{(\alpha )}_{\mathbb T^N\times [0,T^*]}+ |u_0|^{(\alpha +2)}_{\mathbb T^N}\right) + C_2(|u_0|_{\mathbb T^N}+ T^*|u_t|_{\mathbb T^N\times [0,T^*]}). \end{aligned}$$

(A.3)

Taking $T^*$ sufficiently small we can write

$$\begin{aligned} |u|^{(\alpha +2)}_{\mathbb T^N\times [0,T^*]}\le C_3\left( |f|^{(\alpha )}_{\mathbb T^N\times [0,T^*]}+ |u_0|^{(\alpha +2)}_{\mathbb T^N}\right) , \end{aligned}$$

(A.4)

where $C_3$ depends on the coefficients of the equation, on N, T, $\alpha$ and $T^*$ does not depend on $u_0$. We can iterate the estimate (A.4) to cover all the interval [0, T] in $[\frac{T}{T^*}]+1$ steps, thus obtaining (2.8).

The proof of (2.9) is completely analogous. One has to exploit the local estimate in $W^{2,1}_p$ of [20], Eq. (10.12), p. 355. Note that since $R_1^N$ and $R_2^N$ consist of finite copies of $[0,1]^N$, norms on $W_p^{2m, m}(R_i^N \times (0,T))$, $i=1,2$, are multiples (depending on N) of $W_p^{2m, m}(\mathbb T^N \times (0, T))$. $\square$