Quantitative maximal $L^2$-regularity for viscous Hamilton-Jacobi PDEs in 2D and Mean Field Games

TL;DR

This paper establishes quantitative $L^2$-regularity estimates for 2D viscous Hamilton-Jacobi equations and applies these results to prove the existence of classical solutions for certain 2D Mean Field Games systems, also surveying related regularity results.

Contribution

It provides new quantitative Calderón-Zygmund estimates in $W^{2,2}$ for 2D viscous Hamilton-Jacobi equations and uses these to prove classical solutions for 2D Mean Field Games with specific coupling behaviors.

Findings

Quantitative Calderón-Zygmund estimates in $W^{2,2}$ for 2D viscous Hamilton-Jacobi equations.

Existence of classical solutions for stationary 2D Mean Field Games with $m^eta$ coupling.

Survey of regularity results and open problems in viscous Hamilton-Jacobi equations and Mean Field Games.

Abstract

We discuss quantitative Calder\'on-Zygmund estimates in $W^{2,2}$ for 2D viscous Hamilton-Jacobi equations with natural growth in the gradient. We apply the result to obtain the existence of classical solutions for stationary second order Mean Field Games systems in 2D with (defocusing) coupling behaving like $m^\alpha$ for any $\alpha>0$. We also survey on the known results for the regularity of viscous Hamilton-Jacobi equations and second order Mean Field Games and list several open problems.