Convergence of $\partial_{x_j} u(x,t)$ when $u$ converges in $L^2$ norm.

Question

I hope you can help me with this question.

We take $u(x,t)\in L^\infty_{loc}(\mathbb{R},H^1(M))\cap Lip_{loc}(\mathbb{R},L^2(M))$, the derivatives $\partial_{x_j} u $ exist and are continuous, i.e $\partial_{x_j} u(x,t) \rightarrow \partial_{x_j} u_0$, whenever $u\rightarrow u_0$ in $L^2$ norm.

I wrote this argument that because the derivative exist and continuous that the convergence $u\rightarrow u_0$ implies the convergence in the derivatives.

But is this arguement complete, or do i need to add something here?

Thanks in advance.

P.S

$M$ is a compact manifold, and $x_j$ is a variable in the vector $x=(x_1,\ldots , x_n)$. If you have further queries reagrding this question, ask me in the comments.

Answer 1

This will not hold. Take your favorite sequence $\{v_n\}\subset H^1(M)$ and $v \in H^1(M)$ such that $\{v_n\}$ converges towards $v$ in $L^2(M)$, but not in $H^1(M)$.

Now, set $u_n(t,x) = v_n(x)$ and $u(t,x) = v(x)$. This yields $u_n \to u$ in $L^\infty(\mathbb{R};L^2(M))$, but $\partial_{x_j} u_n \not\to \partial_{x_j} u$ in the same space.