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.