Hamilton--Jacobi equations for Wasserstein controlled gradient flows: existence of viscosity solutions

This work is the third part of a program initiated in arXiv:2111.13258,arXiv:2302.06571 aiming at the development of an intrinsic geometricwell-posedness theory for Hamilton-Jacobi equations related to controlledgradient flow problems in metric spaces. In this paper, we finish our analysisin the context of Wasserstein gradient flows with underlying energy functionalsatisfying McCann's condition. More prescisely, we establish that the valuefunction for a linearly controlled gradient flow problem whose running cost isquadratic in the control variable and just continuous in the state variableyields a viscosity solution to the Hamilton-Jacobi equation in terms of twooperators introduced in our former works, acting as rigorous upper and lowerbounds for the formal Hamiltonian at hand. The definition of these operators isdirectly inspired by the Evolutional Variational Inequality formulation ofgradient flows (EVI): one of the main innovations of this work is to introducea controlled version of EVI, which turns out to be crucial in establishingregularity properties, energy and metric bounds along optimzing sequences inthe controlled gradient flow problem that defines the candidate solution.