Existence of Solutions and Selection Problem for Quasi-stationary Contact Mean Field Games

The paper proves existence for (qMFG) via a Schauder fixed point on curves of measures built from the pushforward by the regular Lagrangian flow associated with b(x,t)=∂pH(x,um(t),Dum(t)), using uniform semiconcavity/Lipschitz estimates for viscosity solutions and BV-flow theory; it also establishes a vanishing-contact selection result for (qMFGλ)→(qMFG0) and identifies the limit via the Peierls barrier under a singleton Aubry set (H4). The candidate solution follows the same overall architecture: (i) fix m(·), solve the stationary contact HJ equation, get uniform space semiconcavity/Lipschitz and time regularity controlled by d1, (ii) build the vector field b and solve the continuity equation via Ambrosio–DiPerna–Lions/Ambrosio–Colombo–Figalli regular Lagrangian flows, deriving L∞ and d1-Lipschitz-in-time bounds, and (iii) apply Schauder. For the selection step, the candidate also invokes vanishing-contact/discount convergence and weak KAM identification under (H4). The paper’s technical choices (using BV-flow existence/uniqueness and an inverse-Lipschitz estimate to bound densities) differ mildly from the candidate’s (one-sided Lipschitz and divergence-control/compressibility), but the logic and requirements align. Key components match exactly: the fixed-point map S(m)=Φm♯m0 and its continuity/compactness (Existence proof around (4.1)–Proposition 4.4 and Theorem 2.2), the Lipschitz-in-time estimate for u via the d1-modulus (Proposition 3.3, inequality (3.3)), Lipschitz and semiconcavity of um in x (Proposition 3.2), Lipschitz of c(m) (Lemma 5.2), and the selection theorem and weak KAM identification (Theorem 2.3 and Proposition 5.4). See the fixed-point and flow construction and its stability in the proof of Theorem 2.2 and Proposition 4.4, with flow defined in (4.1), the regularity bounds in Propositions 3.2–3.3 and estimate (3.3), the c(m) bounds/continuity in Lemmas 5.1–5.2, and the selection limit and Peierls barrier representation in Theorem 2.3 and Proposition 5.4 .