  In this paper we construct solutions to Ricci DeTurck flow in four dimensions
on closed manifolds which are instantaneously smooth but whose initial values
$g$ are (possibly) non-smooth Riemannian metrics whose components in smooth
coordinates belong to $W^{2,2}$ and satisfy $ \frac{1}{a}h\leq g\leq a h$ for
some $1<a<\infty$ and some smooth Riemannian metric $h$ on $M$. A Ricci flow
related solution is constructed whose initial value is isometric in a weak
sense to the initial value of the Ricci DeTurck solution. Results for a related
non-compact setting are also presented. Various $L^p$ estimates for Ricci flow,
which we require for some of the main results, are also derived. As an
application we present a possible definition of scalar curvature $\geq k$ for
$W^{2,2}$ metrics $g$ on closed four manifolds which are bounded in the
$L^{\infty}$ sense by $ \frac{1}{a}h\leq g\leq a h$ for some $1<a<\infty$ and
some smooth Riemannian metric $h$ on $M$.
