# Compactness of solutions to parabolic equations (parabolic regularity)

I am working on a Ricci flow $(\mathcal{M} , g(t))$, for which the conjugate heat operator is $\Box ^* := - \partial _t - \Delta + R$, where $R$ is the scalar curvature.

For each $s>0$, I have a solution $\{u_s\}$ of the conjugate heat equation $\Box ^* u_s = 0$ with some 'nice' initial data.

My question is: does anyone have good sources for results like "If we have such-and-such an estimate on the $\{u_s\}$, then we get convergence $\{u_s\} \to u$ smoothly on compact subsets, where $u$ is the fundamental solution of the conjugate heat equation."

It is proving exceedingly difficult to find good parabolic regularity results written down cleanly somewhere!

• What is the meaning of the parameter $s$? What limit are you taking with respect to $s$? – Deane Yang Jan 25 '12 at 20:42
• To be more explicit, the initial data for the $u_s$ are 'generalised Gaussians' on the manifold, i.e. the Euclidean heat kernel where the $|x-y|$ part is replaced with $d_g( \cdot , x_0)$ for some fixed $x_0 \in \mathcal{M}$. I am taking the limit as $s \downarrow 0$. – Michael Coffey Jan 25 '12 at 23:03
• I agree with the last sentence, especially since I recently posted a question about the linear heat equation myself (I had forgotten about this question). – Deane Yang Mar 1 '12 at 9:08
• The book "Global Analysis on Open Manifolds" by Jurgen Eichhorn has been recommended to me. I have not seen a copy yet, but the table of contents looks promising. And surely your advisor Peter Topping can either point you to references or sketch the proofs for you. I'm pretty sure what you want can be established by approximating the equation near a point and using the standard Euclidean heat kernel. The error terms become harmless, if you restrict to a sufficiently small neighborhood of the point. – Deane Yang Mar 1 '12 at 9:15
• I also forgot about this. A great reference is 'Linear and Quasi-Linear Equations of Parabolic Type' by Ladyzhenskaya, Solonnikov and Ural'ceva. It isn't the nicest to read, but the results are very general. It turns out that to answer my question, $L^1$ estimates on the $u_s$ are sufficient, since by a duality argument these can be improved to $L^p$ estimates for some $p$ slightly greater than 1, and then parabolic Calderon-Zygmund-type results will do the rest, setting up an Arzela-Ascoli-type argument. But we get $L^1$ bounds for free by the fact the functions solve the equation. – Michael Coffey Mar 1 '12 at 16:48