One-dimensional, forward-forward mean-field games with congestion

Here, we consider one-dimensional forward-forward mean-field games (MFGs) with congestion, which were introduced to approximate stationary MFGs. We use methods from the theory of conservation laws to examine the qualitative properties of these games. First, by computing Riemann invariants and corresponding invariant regions, we develop a method to prove lower bounds for the density. Next, by combining the lower bound with an entropy function, we prove the existence of global solutions for parabolic forward-forward MFGs. Finally, we construct traveling-wave solutions, which settles in a negative way the convergence problem for forward-forward MFGs. A similar technique gives the existence of time-periodic solutions for non-monotonic MFGs.


Introduction
Mean-field games (MFGs) model competitive interactions in large populations in which the agents' actions depend on statistical information about the population. These games are modeled by a Hamilton-Jacobi equation coupled with a Fokker-Planck equation. An important class of MFGs concerns congestion problems, where the agents' motion is hampered in high-density areas.
In one-dimension, the congestion problem is the system If ε = 0 then (1.1) is a first-order MFG otherwise the system is a parabolic MFG.
For an agent at x ∈ R at time t, the quantity u(x, t) is the value function of a control problem in which the Hamiltonian, H ∈ C 2 , determines the cost and g ∈ C 1 provides a coupling between each agent and the mean field, m. The evolution of m through the Fokker-Planck equation, the second equation in (1.1), also depends on H.
MFGs, introduced in [17], were investigated extensively in the last few years. Numerous results on the existence and uniqueness of solution are now available. For example, the parabolic problem was considered for strong solutions in [11,12,18] and weak solutions in [18,22]. The first-order problem was tackled in [4,5] and weak solutions are obtained. As for the stationary problem, classical and weak solution were sought in [7,8,9,10]. In both standard and stationary problems, the uniqueness of solution is guaranteed by monotonicity of the coupling g. The congestion problem was first addressed in [20] and later, other approaches such as density constraints and nonlinear mobilities were used in [3,21,23]. The existence of smooth solutions for a small terminal time was discussed in [15] and in [16].
Forward-forward MFGs result from reversing the time in the Hamilton-Jacobi equation in (1.1). Here, we focus on the forward-forward MFGs with congestion: with the initial conditions: In [1], the authors propose the forward-forward MFG model and study numerically its convergence to a stationary MFG. This scheme relies on the parabolicity in (1.1) to force the long-time convergence to a stationary solution. In [13], the authors studied parabolic forward-forward MFGs and proved the existence of a solution. Next, in [14] using the entropy method, the convergence for one-dimensional forwardforward MFGs without congestion was proven. Additionally, there, the authors compute entropies for first-order problems and establish connections between these models and certain nonlinear wave equations from elastodynamics.
In [13], the authors proved the existence and regularity for parabolic forwardforward MFGs with subquadratic Hamiltonians. In [19], the authors investigated forward-forward MFGs with logarithmic coupling in the framework of eductive stability. The large-time convergence was studied in [14] for one-dimensional parabolic problems and numerical evidence from [2] and [6] suggests that convergence holds.
The main contributions of this paper are as follows. First, for quadratic Hamiltonians, we convert the forward-forward problem into a system of conservation laws and compute new convex entropies (Lemma 2.1) and Riemann invariants (Lemmas 2.5 and 2.6). These Riemann invariants give lower bounds for the density, m, and, for parabolic MFGs, these bounds combined with an entropy estimate gives the existence of a classical global solution (Theorem 3.6). Finally, by computing traveling-wave solutions, we prove that forward-forward MFGs may fail to converge to a stationary solution. With a similar method, we construct traveling waves for non-monotonic MFGs.

Systems of conservation laws and first-order, forward-forward MFGs
Here, we study the following forward-forward MFG with congestion and a quadratic Hamiltonian: with p ∈ R. Assuming enough regularity, we differentiate the first equation with respect to x and set v = p + u x . We thus obtain the following system of conservation laws 2.1. Existence of convex entropies. First, we construct convex entropies for (2.2). We recall that (η, q) is an entropy/entropy-flux pair for (2.2) if A direct computation shows that η solves (2.3) if and only if it satisfies: In the next lemma, we investigate the existence of entropies and determine conditions under which these entropies are convex.
Proof. By inspection, we see that η as defined in (2.5) satisfies (2.4) if and only if (2.6) holds. Note that Using Sylvester's criterion, that a is even, and that m > 0, we derive (2.7).
Remark 2.2. For 0 < α < 2, the conditions (2.6) and (2.7) hold if a < 0 or a > 1 and with b given by We note that if a > 1 then b < 0. Moreover, we have and, for a > 1, 2.2. Hyperbolicity and Genuinely nonlinearity. Now, we show that (2.2) is a hyperbolic, genuinely nonlinear system of conservation laws. To that end, we compute the Jacobian of F and get More precisely, (2.2) has eigenvalues and (2.14) with corresponding eigenvectors and Furthermore, for 0 < α < 2, the system (2.2) is genuinely nonlinear on Remark 2.4. The set A corresponds to the case where px + u(x) is increasing. Analogous results hold for decreasing functions.
Proof. Simple computations show that DF has eigenvalues given by (2.13) and (2.14). These are distinct if v = 0. Thus, (2.1) is a strictly hyperbolic system of conservation laws if v = 0. Next, we find the right eigenvectors corresponding to λ 1 and λ 2 . Accordingly, we determine r i , i = 1, 2, such that DG T r i = λ i r i . Again, straightforward computations ensure that r 1 , r 2 can be chosen as in (2.15) and (2.16).
Next, we compute Finally, we observe that, for 0 < α < 2, we have The equality in (2.18) holds if and only if v = 0. On the other hand, for any α ∈ R, ∇λ 2 · r 2 0. (2.19) Likewise, the equality in (2.19) holds if and only if v = 0. Thus, for 0 < α < 2, the system is genuinely nonlinear on A.
2.3. Riemann invariants. Now, we compute Riemann invariants for the above system. Later, we show that solutions whose initial values take values in A remain in A. Set and note that A(α) is a positive for all α ∈ R whereas B(α) is negative if and only α > 0. Moreover, In the following Lemmas, we compute Riemann invariants for (2.2).
Lemma 2.5. The family of functions Proof. We recall that z is a Riemann invariant corresponding to r 1 if ∇z · r 1 = 0.

This means that
Thus, a(v) = v s/A(α) and b(m) = m s/2 for s ∈ R. Therefore, To study the convexity of z, we compute its Hessian: First, we observe that the trace of D 2 z(v, m) has the sign of Thus, the trace is positive if only if for all α ∈ R. In view of (2.23), (2.25) and (2.26), if (s, α) ∈ S 0 , then z is convex.

Parabolic forward-forward MFGs
Now, we consider the parabolic forward-forward MFG corresponding to (2.1): where (v 0 , m 0 ) takes values in a compact subset, K, of A, where, as before, Standard PDE theory guarantees that (3.1)-(1.2) has a unique classical solution (v ε , m ε ) on T×[0, T ∞ ) for some 0 < T ∞ ∞. Our goal is to prove that the maximal existence time is T ∞ = ∞.
We recall the Riemann invariants from Lemmas 2.5 and 2.
Proof. First, using (3.1), we get Here, λ 1 and λ 2 are eigenvalues as obtained in Proposition 2.3. Since z and w are convex, we obtain Finally, we use the maximum principle to get that, if w(v 0 , m 0 ) < M and z(v 0 , m 0 ) < M for some M > 0, then the solution, (v ε , m ε ), of (3.1) satisfies Next, we observe that z, w < M , with z and w given in (3.3), implies that m is bounded by below, as can be seen from the level sets of z and m depicted in Figure  2.  Next, we combine the lower bound from the preceding lemma with the entropy from Lemma 2.1 to improve the integrability of v.
By considering the limit a → ∞, we obtain the following corollary.
Proof. The corollary follows by taking a → ∞ and using (2.9). Proof. Using the second equation in (3.1), we have Integrating by parts, we get It thus follows that d dt Next, by Cauchy inequality, We combine (3.11) and (3.12) to obtain d dt Now, for a > 1 and b = b(α, a) as in (2.8), we rewritê Next, we notice that as a → ∞, we have Accordingly, for large enough a, for some µ < 1. Next, we use Morrey's theorem to obtain that Because µ < 1, Young's inequality yields for some C ε,µ . We combine (3.14) with the preceding inequalities to get Finally, we use (3.13) and (3.15) to obtain d dt The estimate (3.10) follows from (3.16).
Finally, we prove our main result, the existence of a global solution for (3.1).
Theorem 3.6. Suppose the initial conditions in (3.2) for (3.1) take values in a compact subset set, K, of A and satisfy w(v 0 , m 0 ) < M and z(v 0 , m 0 ) < M for some M > 0. Then, the maximal existence time T ∞ is T ∞ = +∞.
Proof. Suppose that the maximal existence time, T ∞ , satisfies T ∞ < ∞. First, we notice that by the conservation of mass, we have sup 0 t<T∞ m L 1 = 1.
Thus, using first Lemma 3.5, we obtain that sup 0 t<T∞ m(·, t) L p (T) < C for all p < ∞. Next, using Corollary 3.3, we obtain sup 0 t<T∞ v(·, t) L q (T) < C for all q < ∞. Thus, the solution (v, m) is bounded in L p × L q uniformly in t up to T ∞ . Finally, a simple regularity argument for parabolic equations gives that the solution is classical up T ∞ and, thus, can be continued for t > T ∞ , which contradicts the maximality of T ∞ .

Traveling waves
In this final section, we compute traveling waves for forward-forward MFGs and for MFGs with congestion with an anti-monotonic coupling. For forward-forward MFGs, the existence of traveling waves shows that these PDEs may fail to converge to a stationary solution. Similarly, for MFGs with congestion, the existence of traveling waves indicates that without monotonicity, convergence to a stationary solution may as well not hold. As far as we know, these are the first examples of traveling waves in MFGs.

4.1.
Traveling waves for forward-forward congestion MFGs. We consider the following forward-forward congestion problem: with K > 0. It is straightforward to check that for smooth initial data m 0 , v 0 such that v 0 = cm α 0 with c = ± 2K 3 , we have that m(x, t) = m 0 (x + ct) and v(x, t) = cm α (x, t) solve (4.1).

4.2.
Traveling waves for non-monotonic MFGs with congestion. Now, we consider the following non-monotonic MFG with congestion: where K > 0. When α = 1, then v t + v 2 /(2m) + Km x = 0 m t − v x = 0. Thus, we introduce a potential, q, such that m = q x and v = q t . Accordingly, the first equation in (4.1) becomes − K + q 2 t 2q 2 x q xx + q t q x q xt + q tt = 0, which is a wave-type equation for q.
For the non-monotonic MFG, (4.3) and q such that m = q x and v = q t , we have K − q 2 t 2q 2 x q xx + q t q x q xt + q tt = 0.