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We study the intermediate asymptotic behavior of solutions to the first-order mean field games system with a local coupling, when the initial density is a compactly supported function on the real line, and the coupling is of power type. Addressing a question that was left open in arXiv:2308.00314, we prove that the solutions converge to the self-similar profile. We proceed by analyzing a continuous rescaling of the solution, and identifying an appropriate Lyapunov functional. We identify a critical value for the parameter of the coupling, which determines the qualitative behavior of the functional, and the well-posedness of the infinite horizon system. Accordingly, we also establish, in the subcritical and critical cases, a second convergence result which characterizes the behavior of the full solution as the time horizon approaches infinity. We also prove the corresponding results for the mean field planning problem. A large part of our analysis and methodology apply just as well to arbitrary dimensions. As such, this work is a major step towards settling these questions in the higher-dimensional setting.

We study the behavior of solutions to the first-order mean field games system with a local coupling, when the initial density is a compactly supported function on the real line. Our results show that the solution is smooth in regions where the density is strictly positive, and that the density itself is globally continuous. Additionally, the speed of propagation is determined by the behavior of the cost function near small values of the density. When the coupling is entropic, we demonstrate that the support of the density propagates with infinite speed. On the other hand, for a power-type coupling, we establish finite speed of propagation, leading to the formation of a free boundary. We prove that under a natural non-degeneracy assumption, the free boundary is strictly convex and enjoys $C^{1,1}$ regularity. We also establish sharp estimates on the speed of support propagation and the rate of long time decay for the density. Moreover, the density and the gradient of the value function are both shown to be H\"older continuous up to the free boundary. Our methods are based on the analysis of a new elliptic equation satisfied by the flow of optimal trajectories. The results also apply to mean field planning problems, characterizing the structure of minimizers of a class of optimal transport problems with congestion.

We study the regularity and long time behavior of the one-dimensional, local, first-order mean field games system and the planning problem, assuming a Hamiltonian of superlinear growth, with a non-separated, strictly monotone dependence on the density. We improve upon the existing literature by obtaining two regularity results. The first is the existence of classical solutions without the need to assume blow-up of the cost function near small densities. The second result is the interior smoothness of weak solutions without the need to assume neither blow-up of the cost function nor that the initial density be bounded away from zero. We also characterize the long time behavior of the solutions, proving that they satisfy the turnpike property with an exponential rate of convergence, and that they converge to the solution of the infinite horizon system. Our approach relies on the elliptic structure of the system and displacement convexity estimates. In particular, we apply displacement convexity methods to obtain both global and local a priori lower bounds on the density., Comment: v3: The material on the long time behavior is new. Updated title and abstract

We study the existence of classical solutions to a broad class of local, first order, forward-backward Extended Mean Field Games systems, that includes standard Mean Field Games, Mean Field Games with congestion, and mean field type control problems. We work with a strictly monotone cost that may be fully coupled with the Hamiltonian, which is assumed to have superlinear growth. Following previous work on the standard first order Mean Field Games system, we prove the existence of smooth solutions under a coercivity condition that ensures a positive density of players, assuming a strict form of the uniqueness condition for the system. Our work relies on transforming the problem into a partial differential equation with oblique boundary conditions, which is elliptic precisely under the uniqueness condition., Comment: v2: Fixed typos and misprints, improved readability. To appear in ESAIM Control Optim. Calc. Var

We study the regularity and well-posedness of the local, first-order forward-backward mean field games system, assuming a polynomially growing cost function and a Hamiltonian of quadratic growth. We consider systems and terminal data that are strictly monotone in the density and study two different regimes depending on whether there exists a lower bound for the running cost function. The work relies on a transformation due to P.-L. Lions, which gives rise to an elliptic partial differential equation with oblique boundary conditions, that is strictly elliptic when the coupling is unbounded from below. In this case, we prove that the solution is smooth. When the problem is degenerate elliptic, we obtain existence and uniqueness of weak solutions analogous to those obtained by P. Cardaliaguet and P.J. Graber for the case of a terminal condition that is independent of the density. The weak solutions are shown to arise as viscous limits of classical solutions to strictly elliptic problems., Comment: v3, 29 pages, final version

In this paper the Lagrangian density of a purely kinetic k--essence that models the behavior of dark energy described by four parameterized equations of state proposed by Cooray \& Huterer (Astrophys. J. {\bf 513} L95, 1999), Zhang \& Wu (Mod. Phys. Lett. A {\bf 27} 1250030, 2012), Linder (Phys. Rev. Lett. {\bf 90} 091301, 2003), Efstathiou (Mon. Not. Roy. Astron. Soc. {\bf 310} 842, 2000), and Feng \& Lu (J. Cosmol. Astropart. Phys. {\bf 1111} 034, 2011) has been reconstructed. This reconstruction is performed using the method outlined by de Putter \& Linder (Astropart.\ Phys.\ {\bf 28} 263, 2007), which makes it possible to solve the equations that relate the Lagrangian density of the k--essence with the given equation of state (EoS) numerically. Finally, we discuss the observational constraints for the models based on 586 SNIa data points from the LOSS data set compiled by Ganeshalingam et al. (Mon. Not Roy. Astron. Soc. {\bf 433} 2240, 2013)., Comment: 9 pages, 15 figures. arXiv admin note: text overlap with arXiv:1212.4726 by other authors

Following similar results in arXiv:1301.5934 for flat tori and round spheres, in this paper is presented a proof of the fact that, for "arbitrary" initial conditions $f_0$, the solution $f_t$ at time $t$ of the heat equation on real or complex projective spaces eventually becomes (and remains) a minimal Morse function. Furthermore, it is shown that the solution becomes stable., Comment: 8 pages