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.
Jang et al. (Thu,) studied this question.