Boundedness and convergence

Question

If I know that $\Phi_\varepsilon$ is bounded in $L^{\infty}(\mathbb{R}^{2d})$ and that $\nabla \Phi_\varepsilon$ is bounded in $L^{\infty}(\mathbb{R}^{2d})$, is it true that $\nabla \Phi_\varepsilon \to \nabla \Phi$, where $\Phi$ is the limit of $\Phi_\varepsilon$ as $\varepsilon \to 0$ (in the weak sense)?

Answer 1

$\newcommand\ep\varepsilon$No. E.g., let $\Phi:=0$ and $$\Phi_\ep(x)=\ep e^{-|x|^2}\cos\frac{|x|}\ep$$ for all $x$, where $|x|$ is the Euclidean norm of $x$.

Then $\Phi_\ep\to\Phi$ but $\nabla\Phi_\ep\not\to\nabla\Phi$.