Let $\Omega$ an open subset of $\mathbb{R}^2$ with Lipschitz boundary.
Can I descompose in a unique way any $u\in L^2(0,T;L^2(\Omega))$ such that for all $t\in [0,T]$,
$u(t)=u_1(t)+u_2(t)$
with $\dfrac{d}{dt}u_1(t)=0$ ?
I hope that the descomposition gives $\dfrac{d}{dt}u(t)=\dfrac{d}{dt}u_2(t)$
I think that it's possible using the Ortogonal descoposition theorem on $L^2(\Omega)$, but I'm not sure it is properly.
No, you can not. Let $x \in L^2(\Omega) \setminus \{0\}$ be arbitrary. Then, you have$$u(t) = 0 + u(t) = x + (u(t) - x)$$ and $$\frac{d}{dt} 0 = \frac{d}{dt}x = 0.$$
Note that $d u_1 / dt = 0$ is equivalent to $u_1$ being constant (in time) and there are many constant functions.