Coupled Ricci flow

Question

Let $M$ be a manifold with two metrics $g,g'$ of Ricci curvature $r_g,r_{g'}$. I define a coupled Ricci flow:

$$\frac{\partial g}{\partial t}=-2 r_{g'}$$

$$\frac{\partial g'}{\partial t}=-2r_g$$

Have we solutions for the coupled Ricci flow for short time?