# Classical solutions for parabolic PDE's

I am studying Ricci flow theory and the Ricci flow is not a parabolic equation. But there is some variant of it, called DeTurck-Ricci equation, that happens to be a parabolic PDE. So to argue existence in short time the books I am reading claim: "since the DeTurck-Ricci equation is parabolic, we can ensure existence for short time".

My question is: does the existence of solution is of classic solutions? Which theorem is this claimed? I could not find it in any PDE book I searched.

• If the initial data is $C^2$, then the solution is a smooth and therefore classic solution for $0 \le t < T$ for some $T > 0$. If the initial data is less smooth, then it will be a smooth classic solution at least for $0 < t < T$. – Deane Yang Nov 18 '17 at 0:47
• If you want to consider the variation of Kahler Ricci flow on degeneration of Kahler-Einstein metrics $\pi:X\to \Delta$, then we need to resolve singularities by two times $s$ and $t$, i.e, $$\frac{\partial^2 \omega(s,t)}{\partial s\partial t}=-Ric_{X/\Delta}(s,t)-f(s)\omega(s,t)$$, which is hyperbolic PDE in this case we need to the positivity of initial metric and I am not sure the statement of Deane Yang is true in this case. where $f(s)$ is fiberwise constant. In KRF case we need to run the flow by one time $t$, – user21574 Nov 26 '17 at 23:21
• But in such variant case, we need to rune the flow by two times $s$ and $t$ such that the $s\to 0$ while $t\to\infty$ and $s$ comes from the fibers $\pi^{-1}(s):=X_s$. I think the right flow is hyperbolic type PDE as I wrote in my previous comment. Such flow is the generalized ersion of my question here which I posted 5 years ago mathoverflow.net/questions/106142/… – user21574 Nov 26 '17 at 23:27
• Existence of such solutions correspond to stability of relative tangent sheaf $(T_{X/\Delta})^{**}$ in the sense of Mumford. See the last page slideshare.net/HassanJolany/canonical-metric-on-khler-manifolds – user21574 Nov 26 '17 at 23:30