I'm studying by myself Mean Curvature Flow by Zhu's book and I never did a PDEs course, so I'm trying to learn a little of PDEs as I progress in the study of Mean Curvature Flow and I found some difficulty to apply a PDE's theorem.
Let be $X(x,t)$ a solution of Mean Curvature Flow, $\tilde{X}(y,t)$ a solution of
$$\frac{\partial \tilde{X}}{\partial t} = \triangle_{g(t)} \tilde{X} + v^k \overline{\nabla}_k \tilde{X} \ (*),$$
then $X(x,t) = \tilde{X}(y(x,t),t)$ and the author find
$$\frac{\partial \tilde{X}}{\partial t} = g^{ij} \left\{ \frac{\partial^2 \tilde{X}}{\partial x_i \partial x_j} - \tilde{\Gamma}_{ij}^k \frac{\partial \tilde{X}}{\partial x_k} \right\} \ (**)$$
which is a strictly parabolic equation.
Based on this, the author states that the proposition $2.1$ is a direct consequence of the standard theory of parabolic equations and mention as a reference the book Linear and Quasi-linear Equations of Parabolic Type
$\textbf{Proposition 2.1 (local existence):}$ Let $X_0$ be a smooth hypersurface immersed of $M^n$ into $\mathbb{R}^{n+1}$. Suppose that the second fundamental form $X_0$ bounded, then there exists a positive $\omega$ such that the Cauchy problem
$$\frac{\partial X}{\partial t} = Hn \hspace{1.0 cm} \text{and} \hspace{1.0 cm} X|_{t = 0} = X_0$$
admits a smooth solution $X(\cdot, t)$ on $M^n \times [0, \omega)$. Moreover, if $M^n$ is compact, the solution is unique.
I searched the result on the Olga's book and found it on page $320$, it's the theorem $5.1$. I apologize for not write here all notations, definitions and conditions established on theorem $5.1$, they can all be found on page $2$, $7$, final of the page $317$ and pages $318-320$, which are all available for free viewing here.
$\textbf{Theorem 5.1.}$ Suppose $l > 0$ is a nonintegral number and the coefficients of the operator $\mathcal{L}$ belongs to the class $H^{l,l/2} \left( \overline{D^{(T)}_{n+1}} \right)$, then for any $f \in H^{l,l/2} \left( \overline{D^{(T)}_{n+1}} \right), \ \phi \in H^{l + 2} \left( \overline{E}_n \right)$ problem $(5.2)$ has a unique solution from the class $H^{l/2,l/2+1} \left( \overline{D^{(T)}_{n+1}} \right)$. It satisfies the inequality
$$|u|_{D^{(T)}_{n+1}}^{(l + 2)} \leq c \left( |f|_{D^{(T)}_{n+1}}^{(l)} + |\phi|_{E_n}^{(l + 2)} \right)$$
with the constant not depending on $f$ and $\phi$.
I can see the equation $(**)$ satisfy the hypothesis of theorem $5.1$ if $X$ and all its derivatives are uniformly continuous, but I can't see how the second fundamental form of $X_0$ be bounded imply that $X$ and all its derivatives are uniformly continuous and how the compactness of $M^n$ ensure the uniqueness of the solution of the Mean Curvature Flow, so my doubt is basically how the hypothesis of proposition $2.1$ of Zhu's book ensure that I have the hypothesis of the theorem $5.1$ of Olga's book?
Thanks in advance!
