# Existence, uniqueness, and regularity for linear parabolic PDE on a complete Riemannian manifold

Let $M$ be a smooth manifold with a complete Riemannian metric $g$ and $E$ a smooth vector bundle over $M$ with an inner product and compatible connection $\nabla$. Let $K: E \rightarrow E$ be a smooth bundle map. Then I believe the following is true:

Given any smooth compactly supported section $u_0$ of $E$, there exists a unique solution $u: [0,\infty) \times M \rightarrow E$ of the linear heat equation $$\partial_t u = g^{ij}\nabla_i\nabla_j u + Ku$$ such that $u$ is smooth for $t > 0$ and the $L^p$ norm of $u(t,\cdot)$ is bounded for each $t \ge 0$ and $1 \le p \le \infty$.

If I'm wrong, please explain. Otherwise, is there a reference for this somewhere? If not the exact theorem, a proof of a similar theorem that can be adapted to a proof of this?

ADDED: I'm mostly interested in proving the existence statement and preferably using a standard PDE approach. It appears to me that there is a straightforward argument starting by approximating the equation by the standard constant coefficient heat equation on a sufficiently small co-ordinate chart and patching together local solutions to the constant coefficient equation to get a global section. You then use a priori estimates to show that the global section is an approximate solution to the original system. This can then be iterated into an exact solution. But the details are a bit tedious, so it would be nice if I could find it all written down somewhere. And it would be nice to be clear about what, if any, assumptions on the Riemannian metric are needed to make this all work.

ADDED: I'm being a little slow here. Since everything is linear and homogeneous, it suffices to prove existence for initial data supported on a sufficiently small co-ordinate chart. If you start with compactly supported initial data, then you can always write it as a finite sum, where each term is supported in a sufficiently small open set.

ADDED: I took another quick look at a partition of unity argument for reducing the question to solving a parabolic PDE on an open set in $R^n$. It looks like this really does work pretty easily. But it still seems like something that someone should write up carefully and publish in a book somewhere.

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Maybe the thesis of Alexander Polden can help. You can find the part you need in this article (last section) pubman.mpdl.mpg.de/pubman/item/escidoc:152741:1/component/… – Chih-Wei Chen Feb 2 '12 at 3:29

I've played with such things 15 years ago. Here's what I remember...

Sure you need some semi-boundedness condition on K. Uniqueness should then follow via semigroup domination (aka Kato's inequality) from scalar case.

I. Shigekawa, L^p Contraction Semigroups for Vector Valued Functions, J. Funct. Analysis 147 (1997), 69-108

Shigekawa had rediscovered Barry Simon's semigroup domination criterion (i.e. Kato's inequality) and generalized it to the vector valued things.

R.S. Strichartz, Analysis of the Laplacian on the Complete Riemannian Manifold, J. Funct. Analysis 52 (1983), 48-79. Scalar case: Theorem 3.5 - but has uniqueness only for 1< p<∞.

(Some details in Strichartz' proof(s) for vector valued Laplacians not optimal (classic problems with doing tensor calculus and Stokes...) not even in the follow-up paper "L^p contractive projections and the heat semigroup for differential forms" J. Funct. Analysis, 65 (1985), 348-357)

Perhaps look also for El Maati Ouhabaz ca. 1999. He's written a book, "Analysis of Heat Equations on Domains" (2004) but I never got hold of it.

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Thanks but I'm mostly interested in the existence statement. Do you have any suggestions for that? – Deane Yang Jan 30 '12 at 11:52
Back then I couldn't find any "nice" literature, so I fancied (but never worked out) something like that: 1. Use the L^2 spectral theorem. 2. Assume a compact manifold (glue small enough open subset of your general manifold in a compact one). You get a smooth map from positive reals into L^2 of course, but also into Sobolev spaces (expand powers of the Laplacian left to the exponential into powers of covariant derivative). No need for constant coefficients or a Friedrichs mollifier. But you need the Friedrichs inequality. My old notes recommend a book by E.B.Davies, Heat kernels and spct thry – Martin Gisser Feb 8 '12 at 1:12
Sorry, I'm far away from the library and won't be able to recall much more. This stuff had bugged me quite a bit - but I had a very individual approach to things (e.g. hating coordinates and Christoffel stuff and Warner's GTM94 book). I felt encouraged by a quote from Bourguignon's Weitzenböck paper: He said, if you encounter variable coefficients, go for geometric tools. (That's contrary to your ansatz in 1st ADDED) --- What I possibly liked more was J.Roe: "Elliptic Operators, topology and asymptotic methods", Pitman Res. Notes 179 (1988) --- I'm extremely curious what you will find out! – Martin Gisser Feb 8 '12 at 1:31

There is a paper http://ir.library.osaka-u.ac.jp/metadb/up/LIBOJMK01/ojm32_02_04.pdf by K. Sturm considering parabolic equations in a still greater generality, in the language of Dirichlet forms. It contains a long list of references, some of which are related to your question.

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