From the book Geometric Analysis by Peter Li, we have the gradient estimate of heat equation as follows:
Theorem Let $M^m$ be a complete manifold with boundary. Assume that $p \in M$ and $\rho>0$ so that the geodesic ball $B_p(4 \rho)$ does not intersect the boundary of $M$. Suppose the Ricci curvature of $M$ on $B_p(4 \rho)$ is bounded from below by $$ \mathcal{R}_{i j} \geq-(m-1) R $$ for some constant $R \geq 0$. If $f(x, t)$ is a positive solution of the equation $$ \left(\Delta-\frac{\partial}{\partial t}\right) f(x, t)=0 $$ on $M \times[0, T]$, then for any $1<\alpha$, the function $g(x, t)=\log f(x, t)$ must satisfy the estimate $$ |\nabla g|^2-\alpha g_t \leq \frac{m}{2} \alpha^2 t^{-1}+C_1 \alpha^2(\alpha-1)^{-1}\left(\rho^{-2}+R\right) $$ on $B_p(2 \rho) \times(0, T]$, where $C_1$ is a constant depending only on $m$.
It means that $$ |\nabla f|^2-\alpha f f_t \leq C f^2 \quad \text { on } \quad B_p(\rho) $$ Thus we have
If $\phi$ is a nonnegative cutoff function supported on $B_p(\rho)$, then $$ \begin{aligned} \int_M \phi^2|\nabla f|^2 & \leq \alpha \int_M \phi^2 f f_t+C \int_M \phi^2 f^2 \\ & \leq \alpha \int_M \phi^2 f \Delta f+C \int_M \phi^2 f^2 \\ & \leq-\alpha \int_M \phi^2|\nabla f|^2-2 \alpha \int_M \phi f\langle\nabla \phi, \nabla f\rangle+C \int_M \phi^2 f^2 . \end{aligned} $$ Applying the Schwarz inequality $$ -2 \int_M \phi f\langle\nabla \phi, \nabla f\rangle \leq \int_M \phi^2|\nabla f|^2+\int_M|\nabla \phi|^2 f^2, $$ we obtain the estimate $$ \begin{aligned} \int_{B_p(\rho / 2)}|\nabla f|^2 & \leq C \int_M \phi^2 f^2+\alpha \int_M|\nabla \phi|^2 f^2 \\ & \leq C_1 . \end{aligned} $$ by choosing $\phi=1$ on $B_p(\rho / 2)$.
Therefore from the uniform $L^\infty$ bound of solutions of the heat equation, we can only get the uniform $L^2$ bound of their derivatives.
However, in many papers, given the uniform $L^\infty$ bound of solutions of the heat equation on compact subsets, they claimed that after passing to subsequence, the solutions and their derivatives both converge uniformly. Can anyone help to explain this? Thanks in advance.