In Evans' PDE bok (Second Edition) p.384 Theorem 5, it is stated essentially that the weak solution $u(x,t) : U \times [0,T] \to \mathbb{R}^n$ of the parabolic equation
\begin{equation}
\partial_t u -\Delta u=f \text{ with } u(\cdot,t)=0 \text{ on } \partial U \text{ and } u(\cdot,0)=g
\end{equation}
for $f \in L^2([0,T] \times U)$ and $g \in H^1_0(U)$ has $2$ more order spatial regularity than $f$. That is, $u
\in L^2([0,T] \times H^2(U))$.
However, in this theorem, the "spatial domain" $U$ is only assumed to be an open bounded subset of some Euclidean space.
I wonder if the same result still holds for a compact smooth manifold without boundary.
For example, if we set $U =\mathbb{T}^k:= [\mathbb{R}/\mathbb{Z}]^k$ for some $k \in \mathbb{N}$, does the theorem holds valid exactly in the same form?
Could anyone please clarify for me, or provide any reference?
Best Answer
In general yes, because these regularity results are local in nature (as is evident from the proofs). Once you have a weak solution $u$, you can map small neighbourhoods of your manifold to a Euclidean domain $U$ via local coordinates. If your compact manifold $M$ is smooth, then ellipticity and smoothness of coefficients should be preserved under this diffeomorphism, and you can apply the standard theory in the coordinate space. Note that the Laplacian needs to replaced by a suitable geometric counterpart to make sense, such as the Laplace-Beltrami operator on Riemannian manifolds.
For the flat torus in particular, you're just looking at 1-periodic solutions to the heat equation, so your diffeomorphism is simply given by local restrictions of this periodisation map.
While I'm not aware of specific references that prove this in detail, I would recommend knowing this localisation argument as a general principle, where you can reproduce the details as you need them. This way you'll see that this not only applies to $L^2$-$H^2$ estimates, but other estimates like Schauder, Calderón-Zygmund, etc.