My question is the following:
What is the minimum regularity for a continuous loop $\gamma: S^1 \rightarrow M$ in a Riemannian manifold $M$ to have short-time existence for the harmonic map flow in a suitably sense?
It is well-known that if you start with a $C^1$ loop $\gamma$, then the harmonic map flow $$ \frac{\mathrm{d}}{\mathrm{d} t} \gamma_t = \frac{\nabla}{\partial \theta} \frac{\partial}{\partial \theta}{\gamma}_t(\theta), ~~~~~ \gamma_0(\theta) = \gamma(\theta)$$ exists in the classical sense, and $\gamma_t$ is smooth for $t \rightarrow 0$.
It is also clear that if the target is $\mathbb{R}^n$, then the differential equation is linear, so it makes sense to say when a solution solves the equation in the distributional sense, and the result is that solutions to any distributional initial value exist and become smooth immediately. In the general case that $M$ is curved, the equation will be non-linear, and it is harder to say what weak solutions are.
So precisely: Is there a good notion of a weak solution of the harmonic map flow for, say, $C^\alpha$ functions $\alpha > 0$? If so, do solutions exist for short time and become immediately smooth? And do solutions depend continuously on initial data?
