Conformal heat flow of maps under Yamabe flow of metrics

We study the gradient flow of the sum of the Dirichlet energy of a map from a low-dimensional closed manifold to an arbitrary-dimensional Riemannian manifold, and the normalized Yamabe energy on the domain. We show that a smooth solution to the system exists for short time, and that, with small initial error data from an equilibrium, the solution exists and is stable for all time.