Hi I have the next claim which I would like to find a proof of it.

I have a sequence of functions $u_\epsilon(t,x) \in H^1(M)$ where $M$ is a compact manifold, and $u_\epsilon \in L^\infty(I,L^2(M))\cap Lip(I,L^2(M))$ where $I$ is some interval that includes zero in it.

The claim is that $\partial_t u_\epsilon$ is bounded in $L^\infty(I,L^2(M))$, why is that necessarily true?

Thanks in advance.

Hi I have the next claim which I would like to find a proof of it.

I have a sequence of functions $u_\epsilon(t,x) \in H^1(M)$ where $M$ is a compact manifold, and $u_\epsilon \in L^\infty(I,H^1(M))\cap Lip(I,L^2(M))$ where $I$ is some interval that includes zero in it.

The claim is that $\partial_t u_\epsilon$ is bounded in $L^\infty(I,L^2(M))$, why is that necessarily true?

Thanks in advance.