Two notions of distributional derivative on domains $Q=(0,T)\times \Omega$

Question

Let $Q=(0,T)\times (0,1)$, and $u\in L^2(Q)$. We can define two notions of distribuitional derivative.

1. The usual one: $D_tu,D_xu\in \mathcal{D}'(Q)$ defined by

$\langle\langle D_tu,\phi\rangle\rangle:=-((u,\phi_t)):=\iint_Qu(t,x)\phi_t(t,x)dxdt$ and $\langle\langle D_xu,\phi\rangle\rangle:=-((u,\phi_x))$ for all $\phi\in \mathcal{D}(Q)$.

2. The vector one: Considering $u\in L^2(0,T;L^2(0,1))$ we can define $u_t\in \mathcal{D}(0,T;L^2(0,1)) $ as

$\langle u_t,\varphi\rangle:= -(u(\cdot,x),\varphi')_{L^2(0,T)}=\int_0^Tu(t,x)\varphi'(t)dt\in L^2(0,1)$ for all $\varphi\in \mathcal{D}(0,T).$

And, we can also define $u_x:[0,T]\to H^{-1}(0,1)$ as

$\langle u_x(t),\psi\rangle:=-(u(t,\cdot),\psi')_{L^2(0,1)}=\int_0^1 u(t,x)\psi'(x)dx$ for all $\psi \in \mathcal{D}(0,1).$

I would like to know if there is a context where those notions are the same, that is, $D_tu=u_t$ and $D_xu=u_x$. If possible, I would to like to have a good reference.

Answer 1

I have found the answer. That two notions coincide because the subspace of $C^\infty_c(Q)$ generated by the product of functions $\varphi\psi$, where $\varphi\in \mathcal{D}(0,T)$ and $\psi\in \mathcal{D}(0,1)$ is dense in $\mathcal{D}(Q)$. So, it easy to see that the two notions coincide when $\phi(t,x)=\varphi(t)\phi(x)$. By density we can define $u_t$ an $u_x$ as distribution in $\mathcal{D}(Q)$.

That result about density can be found in Theorem 4.3.1 of

Friedlander, F. G., Joshi, M. S., Joshi, M., and Joshi, M. C. Introduction to the Theory of Distributions. Cambridge University Press, 1998

.