The scalar curvature flow

Question

Let $(M,g)$ be a riemannian manifold with Levi-Civita connection $\nabla$. I define the scalar curvature flow over the metric $g$:

$$\frac{\partial g}{\partial t}(X,Y)= (XY-\nabla_X Y)r$$

where $r$ is the scalar curvature of $g$.

Have we solutions of the scalar curvature flow for short time?