Maximal $L^q$-regularity for parabolic Hamilton-Jacobi equations and applications to Mean Field Games

TL;DR

This paper establishes maximal L^q-regularity for viscous Hamilton-Jacobi equations with unbounded terms, providing new analytical tools and applying them to solve classical solutions in Mean Field Games with unbounded couplings.

Contribution

It introduces a novel duality-based approach to achieve maximal regularity for parabolic Hamilton-Jacobi equations with unbounded right-hand sides and applies these results to Mean Field Games.

Findings

Proved maximal L^q-regularity for viscous Hamilton-Jacobi equations.

Developed new integral and Hölder estimates for these equations.

Applied the theory to establish existence of classical solutions in Mean Field Games.

Abstract

In this paper we investigate maximal $L^q$-regularity for time-dependent viscous Hamilton-Jacobi equations with unbounded right-hand side and superlinear growth in the gradient. Our approach is based on the interplay between new integral and H\"older estimates, interpolation inequalities, and parabolic regularity for linear equations. These estimates are obtained via a duality method \`a la Evans. This sheds new light on a parabolic counterpart of a conjecture by P.-L. Lions on maximal regularity for Hamilton-Jacobi equations, recently addressed in the stationary framework by the authors. Finally, applications to the existence problem of classical solutions to Mean Field Games systems with unbounded local couplings are provided.