A flow on exterior forms

Question

Let $(M,g)$ be a riemannian manifold with Levi-Civita connection $\nabla^{LC}$ acting on the tensor algebra. The codifferential on exterior forms is $\delta=\star \circ d \circ \star$, with $\star$ the Hodge operator. Then, I define a flow on $\omega \in \Lambda^2(TM)$:

$$\frac{\partial \omega}{\partial t}=\nabla^{LC}_{(\delta \omega)^*}\omega$$

Can we find solutions of this flow for small times?