I have the following problem:
$iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm $\|f\|_{L^2(x,t)}$, i.e., there is a $C$ so that $\|u\|_{L^2(x,t)}\leq C\|f\|_{L^2(x,t)}$. Is is possible to control the norm of $\frac{\partial u}{ \partial x}$ , i.e., $\|u_{x}\|_{L^2(x,t)}$ by $\|f\|_{L^2(x,t)}$? If not, why? I think the use of weighted $L^2$ inner products will help, but am not sure.