Well-posedness of mean field games master equations involving non-separable local Hamiltonians (2023)
Journal Article
Ambrose, D. M., & Mészáros, A. R. (2023). Well-posedness of mean field games master equations involving non-separable local Hamiltonians. Transactions of the American Mathematical Society, 376(4), 2481-2523. https://doi.org/10.1090/tran/8760

In this paper we construct short time classical solutions to a class of master equations in the presence of non-degenerate individual noise arising in the theory of mean field games. The considered Hamiltonians are non-separable and local functions o... Read More about Well-posedness of mean field games master equations involving non-separable local Hamiltonians.

Mean field games master equations with nonseparable Hamiltonians and displacement monotonicity (2022)
Journal Article
Gangbo, W., Mészáros, A. R., Mou, C., & Zhang, J. (2022). Mean field games master equations with nonseparable Hamiltonians and displacement monotonicity. Annals of Probability, 50(6), 2178-2217. https://doi.org/10.1214/22-aop1580

A variational approach to first order kinetic Mean Field Games with local couplings (2022)
Journal Article
Griffin-Pickering, M., & Mészáros, A. R. (2022). A variational approach to first order kinetic Mean Field Games with local couplings. Communications in Partial Differential Equations, 47(10), 1945-2022. https://doi.org/10.1080/03605302.2022.2101003

First order kinetic mean field games formally describe the Nash equilibria of deterministic differential games where agents control their acceleration, asymptotically in the limit as the number of agents tends to infinity. The known results for the w... Read More about A variational approach to first order kinetic Mean Field Games with local couplings.

Global Well‐Posedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games (2022)
Journal Article
Gangbo, W., & Mészáros, A. R. (2022). Global Well‐Posedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games. Communications on Pure and Applied Mathematics, 75(12), 2685-2801. https://doi.org/10.1002/cpa.22069

This manuscript constructs global in time solutions to master equations for potential mean field games. The study concerns a class of Lagrangians and initial data functions that are displacement convex, and so this property may be in dichotomy with t... Read More about Global Well‐Posedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games.

Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients (2021)
Journal Article
Kwon, D., & Mészáros, A. R. (2021). Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients. Journal of the London Mathematical Society, 104(2), 688-746. https://doi.org/10.1112/jlms.12444

This paper is devoted to the study of some nonlinear parabolic equations with discontinuous diffusion intensities. Such problems appear naturally in physical and biological models. Our analysis is based on variational techniques and in particular on... Read More about Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients.

Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport (2020)
Journal Article
Jacobs, M., Kim, I., & Mészáros, A. R. (2020). Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport. Archive for Rational Mechanics and Analysis, 239(1), 389-430. https://doi.org/10.1007/s00205-020-01579-3

Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedoḡlu–Otto and Laux–Otto), we introduce a novel framework to approximate solutions of the Muskat problem with surface tension. Our approach is based... Read More about Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport.

The planning problem in mean field games as regularized mass transport (2019)
Journal Article
Graber, P. J., Mészáros, A. R., Silva, F. J., & Tonon, D. (2019). The planning problem in mean field games as regularized mass transport. Calculus of Variations and Partial Differential Equations, 58(3), Article 115. https://doi.org/10.1007/s00526-019-1561-9

In this paper, using variational approaches, we investigate the first order planning problem arising in the theory of mean field games. We show the existence and uniqueness of weak solutions of the problem in the case of a large class of Hamiltonians... Read More about The planning problem in mean field games as regularized mass transport.

On nonlinear cross-diffusion systems: an optimal transport approach (2018)
Journal Article
Kim, I., & Mészáros, A. R. (2018). On nonlinear cross-diffusion systems: an optimal transport approach. Calculus of Variations and Partial Differential Equations, 57(3), Article 79. https://doi.org/10.1007/s00526-018-1351-9

We study a nonlinear, degenerate cross-diffusion model which involves two densities with two different drift velocities. A general framework is introduced based on its gradient flow structure in Wasserstein space to derive a notion of discrete-time s... Read More about On nonlinear cross-diffusion systems: an optimal transport approach.

On the Variational Formulation of Some Stationary Second-Order Mean Field Games Systems (2018)
Journal Article
Mészáros, A. R., & Silva, F. J. (2018). On the Variational Formulation of Some Stationary Second-Order Mean Field Games Systems. SIAM Journal on Mathematical Analysis, 50(1), 1255-1277. https://doi.org/10.1137/17m1125960

We consider the variational approach to prove the existence of solutions of second-order stationary Mean Field Games systems on a bounded domain $\Omega\subseteq {\mathbb R}^{d}$ with Neumann boundary conditions and with and without density constrain... Read More about On the Variational Formulation of Some Stationary Second-Order Mean Field Games Systems.

Sobolev regularity for first order mean field games (2018)
Journal Article
Jameson Graber, P., & Mészáros, A. R. (2018). Sobolev regularity for first order mean field games. Annales de l'Institut Henri Poincaré C, 35(6), 1557-1576. https://doi.org/10.1016/j.anihpc.2018.01.002

In this paper we obtain Sobolev estimates for weak solutions of first order variational Mean Field Game systems with coupling terms that are local functions of the density variable. Under some coercivity conditions on the coupling, we obtain first or... Read More about Sobolev regularity for first order mean field games.

First Order Mean Field Games with Density Constraints: Pressure Equals Price (2016)
Journal Article
Cardaliaguet, P., Mészáros, A. R., & Santambrogio, F. (2016). First Order Mean Field Games with Density Constraints: Pressure Equals Price. SIAM Journal on Control and Optimization, 54(5), 2672-2709. https://doi.org/10.1137/15m1029849

In this paper we study mean field game systems under density constraints as optimality conditions of two optimization problems in duality. A weak solution of the system contains an extra term, an additional price imposed on the saturated zones. We sh... Read More about First Order Mean Field Games with Density Constraints: Pressure Equals Price.

Uniqueness issues for evolution equations with density constraints (2016)
Journal Article
Di Marino, S., & Mészáros, A. R. (2016). Uniqueness issues for evolution equations with density constraints. Mathematical Models and Methods in Applied Sciences, 26(09), 1761-1783. https://doi.org/10.1142/s0218202516500445

In this paper, we present some basic uniqueness results for evolution equations under density constraints. First, we develop a rigorous proof of a well-known result (among specialists) in the case where the spontaneous velocity field satisfies a mono... Read More about Uniqueness issues for evolution equations with density constraints.

Advection-diffusion equations with density constraints (2016)
Journal Article
Mészáros, A. R., & Santambrogio, F. (2016). Advection-diffusion equations with density constraints. Analysis & PDE, 9(3), 615-644. https://doi.org/10.2140/apde.2016.9.615

In the spirit of the macroscopic crowd motion models with hard congestion (i.e., a strong density constraint ρ≤1) introduced by Maury et al. some years ago, we analyze a variant of the same models where diffusion of the agents is also taken into acco... Read More about Advection-diffusion equations with density constraints.

BV Estimates in Optimal Transportation and Applications (2016)
Journal Article
De Philippis, G., Mészáros, A. R., Santambrogio, F., & Velichkov, B. (2016). BV Estimates in Optimal Transportation and Applications. Archive for Rational Mechanics and Analysis, 219(2), 829-860. https://doi.org/10.1007/s00205-015-0909-3

In this paper we study the BV regularity for solutions of certain variational problems in Optimal Transportation. We prove that the Wasserstein projection of a measure with BV density on the set of measures with density bounded by a given BV function... Read More about BV Estimates in Optimal Transportation and Applications.

A variational approach to second order mean field games with density constraints: The stationary case (2015)
Journal Article
Mészáros, A. R., & Silva, F. J. (2015). A variational approach to second order mean field games with density constraints: The stationary case. Journal de Mathématiques Pures et Appliquées, 104(6), 1135-1159. https://doi.org/10.1016/j.matpur.2015.07.008

In this paper we study second order stationary Mean Field Game systems under density constraints on a bounded domain . We show the existence of weak solutions for power-like Hamiltonians with arbitrary order of growth. Our strategy is a variational o... Read More about A variational approach to second order mean field games with density constraints: The stationary case.