$\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, consider the heat equation on $\Omega$ with Dirichlet boundary condition \begin{cases} \partial_t u -\Delta u=0 & (t,x)\in (0,\infty)\times \Omega\\ u(0,x)=u_0(x) & (t,x)\in \{0\}\times \Omega \\ u(t,x)=0 & (t,x)\in (0,\infty)\times \partial \Omega \end{cases} By semigroup theory, Laplacian operator $\Delta:H^1_0(\Omega)\cap H^2(\Omega)\to L^2(\Omega)$, generates a bounded analytic semigroup $E(t)=e^{t\Delta}$ on $L^2(\Omega)$. Thus for $u_0\in L^2(\Omega)$, there is solution $u(t,\cdot)=E(t)u_0\in H^1_0(\Omega)\cap H^2(\Omega), \forall t>0$ satisfying the equation and $\|u(t,\cdot)-u_0\|_{L^2(\Omega)}\to 0$ when $t\to 0$, with $\|u(t,\cdot)\|_{L^2(\Omega)}\leq C\|u_0\|_{L^2(\Omega)}$.
My question is, if we moreover assume that $u_0\in W^{1,\infty}(\Omega)\cap H^1_0(\Omega)$, is there exists following gradient estimation:
$$ \|u(t,\cdot)\|_{W^{1,\infty}(\Omega)}\leq C\|u_0\|_{W^{1,\infty}(\Omega)} \quad \forall t>0, $$ where $C$ is a constant independent of $u_0$ and $t$.