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.

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Acknowledgements

Marco Cirant and Paola Mannucci are members of GNAMPA-INdAM, and were partially supported by the research project of the University of Padova “Mean-Field Games and Nonlinear PDEs” and by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games”.

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Appendix

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 \(W_p^{2m, m}(\mathbb T^N \times (0, T))\). \(\square \)

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Cirant, M., Gianni, R. & Mannucci, P. Short-Time Existence for a General Backward–Forward Parabolic System Arising from Mean-Field Games. Dyn Games Appl 10, 100–119 (2020). https://doi.org/10.1007/s13235-019-00311-5

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Keywords

  • Parabolic equations
  • Backward–forward system
  • Mean-field games
  • Hamilton–Jacobi
  • Fokker–Planck
  • Congestion problems

Mathematics Subject Classification

  • 35K40
  • 35K61
  • 49N90