Inserted: 31 jul 2020
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.
Keywords: adjoint method, Mean-Field Games, H\"older regularity, Riccati equation, Maximal $L^q$-regularity,, Hamilton-Jacobi equations with unbounded right-hand side, Khardar-Parisi-Zhang equation