# Infinite differentiability for a solution of the general linear parabolic pde of second order

I'm studying by myself the the chapter of second order parabolic linear equations by Evan's book, which focus in solve

$$(11) \ \begin{cases} \begin{eqnarray*} u_t + Lu &=& f \ \text{in} \ U_T\\ u &=& 0 \ \text{on} \ \partial U \times [0,T]\\ u &=& g \ \text{on} \ U \times \{ t = 0 \} \end{eqnarray*} \end{cases}$$

I'm trying understand how the theorem $$7$$ on page $$388$$ is a direct consequence of the theorem $$6$$ on page $$386$$. Specifically, I'm trying understand how the following theorem:

$$\textbf{Theorem 7.}$$ (Infinite differentiability). Assume

$$g \in \mathcal{C}^{\infty}(\overline{U}), f \in \mathcal{C}^{\infty}(\overline{U}_T),$$

and the $$m^{th}$$-order compatibility conditions hold for $$m = 0, 1, \cdots$$.

Then the parabolic initial/boundary value-problem $$(11)$$ has an unique solution

$$u \in \mathcal{C}^{\infty}(\overline{U}_T).$$

is a direct consequence of this theorem:

$$\textbf{Theorem 6.}$$ (Higher Regularity). Assume

$$\begin{cases} g \in H^{2m+1}(U),\\ \frac{d^kf}{dt^k} \in L^2(0,T;H^{2m-2k}(U)) \ (k = 0, \cdots, m). \end{cases}$$

Suppose also that the following $$m^{th}$$-order compatibility conditions hold

$$\begin{cases} g_0 := g \in H^1_0(U), g_1 := f(0) - L g_0 \in H^1_0(U),\\ \cdots, g_m := \frac{d^{m-1}f}{dt^{m-1}}(\cdot, 0) - Lg_{m-1} \in H^1_0(U). \end{cases}$$

Then

$$\frac{d^k\textbf{u}}{dt^k} \in L^2 (0,T;H^{2m+2-2k}(U)) \ (k = 0, \cdots, m+1),$$

and we have the estimate

$$(55) \ \sum_{k=0}^{m+1} \left| \left| \frac{d^k\textbf{u}}{dt^k} \right| \right|_{H^{2m+2-2k}(U)} \leq C \left( \sum_{k=0}^{m} \left| \left| \frac{d^k\textbf{f}}{dt^k} \right| \right|_{L^2(0,T;H^{2m-2k}(U))} + ||g||_{H^{2m+1}(U)} \right),$$

the constant $$C$$ depending only on $$m, U, T$$ and the coefficients of $$L$$.

Evans said simply that the theorem $$7$$ follows from the theorem $$6$$ applying this theorem for $$m = 0, 1, 2, \cdots$$, but I didn't understand how the bounds for the higher derivatives of $$\textbf{u}$$ with respect for the norm of $$L^2(0,T;H^{2m+2-2k}(U))$$ imply that these same higher derivatives are bounded with respect to the uniform norm over $$\overline{U}_T$$, which imply that $$u \in \mathcal{C}^{\infty}(\overline{U}_T)$$. I would like to understand it.

P.S.: just to make the question self-contained, I will introduce some definitions used on theorems $$6$$ and $$7$$.

$$\textbf{Definition.}$$ Let be $$X$$ a real Banach space with norm $$|| \ ||$$. The space $$L^p(0,T;X)$$ consists of all strongly measurable functions $$\textbf{u}: [0,T] \longrightarrow X$$ with

$$(i) \ ||\textbf{u}||_{L^p(0,T;X)} := \left( \int_0^T ||\textbf{u}(t)||^p \right)^{\frac{1}{p}} < \infty$$

for $$1 \leq p < \infty$$.

$$\textbf{Definition.}$$ Let be $$U$$ an open set of $$\mathbb{R}^n$$, then $$H^k(U) := W^{k,2}(U)$$.

• Without having lokked at the details, this is probably an application of the Sobolev embedding $H^k\hookrightarrow C^l$ for $k$ sufficiently large. – MaoWao Mar 3 at 11:49
• Can you suggest a reference for this embedding please? I looked for this result on Evan's book, but I didn't find. – George Mar 4 at 13:42
• I don't have Evans's book in front of me, but I'm pretty sure this embedding is mentioned in Section 5.6 (or somewhere nearby). Also I'm only talking about the scalar-valued Sobolev spaces, you have to combine the bounds for the time and space variable to get smoothness of the function here. – MaoWao Mar 4 at 13:55
• The only embeddings that I saw in Evan's book is Gagliardo-Nirenberg-Sobolev inequality, Morrey's inequality and the general Sobolev inequalities, which can be read here. The Wikipedia's article comments on the final of the section "Sobolev embedding theorem" that exists an embedding $W^{k,p}(\mathbb{R}^n) \subset \mathcal{C}^{r,\alpha}(\mathbb{R}^n)$, which is a direct consequence of Morrey's inequality according the article, but I can't see this since Morrey's inequality suppose that $u \in \mathcal{C}^1$ and I don't know this in my case. – George Mar 4 at 16:33
• Furthermore, I found a result that states that $\mathcal{C}^{k,\gamma}(\Omega) \hookrightarrow \mathcal{C}^k(\overline{\Omega})$ (see the "Teorema $3.17$" here on page $56$ of the PDF). If I understood well, it's always valid that $W^{k,p}(\mathbb{R}^n) \hookrightarrow \mathcal{C}^{r,\alpha}(\mathbb{R}^n) \hookrightarrow \mathcal{C}^r(\mathbb{R}^n)$, isn't it? – George Mar 4 at 16:43