My question is about a passage in the paper 'On the existence of the pressure for solutions of the variational Navier–Stokes equations' by J.Simon. (It can be found easily via Google). It says the following on page 4 and 5:
We are now in position to check that, for such a right-hand side, it is not possible to associate a pressure to the velocity given by the weak equation in order to satisfy the Navier–Stokes equation. Let us recall, for $d \leq 4$, an existence result of a weak solution given by J.–L. Lions in [5], see Theorem 6.1 p. 69, Corollary 6.2 and Footnote (2) p. 74 (see also [10], Theorem 3.1 p. 282 together with the equivalence of problems 3.1 and 3.2 stated p. 282). Given $$f ∈L^2\big((0,T);V′\big), u_0 \in H$$ there exists $$u ∈ L^2 \big((0, T ); V \big) ∩ L^\infty \big((0, T ); H \big) ∩ C \big([0, T ]; H_\text{weak}\big) \tag6$$ such that $u(0) = u_0$ and $$⟨∂_t u − ν∆u + (u \cdot ∇)u, φ⟩_{H^{−1}(Ω)d×H_0^1(Ω)} = ⟨f, φ⟩_{V'×V}, ~∀φ ∈ V. \tag7$$ The terms in the left-hand side are defined in the following space: $$∂_t u∈H^{−1}\big((0,T);H\big), (u\cdot∇)u∈L^1\big((0,T);L^{4/3}(Ω)\big), ∆u ∈ L^2\big((0, T ); H^{−1}(Ω)\big).\tag8$$
I checked his reference [10] (Temam: Navier-Stokes Equations) which says that the weak time derivative of $u$ is in $L^1(0,T;V')$ and can be improved to $L^{4/3}(0,T;V')$ in three dimensions. Now I don't get how Simon can say that $\partial_t u \in H^{-1}(0,T;H)$. It seems like he is saying since $u\in L^2(0,T;H)$ that for the weak derivative one takes one step from $W^{0,2}(0,T;H)$ to $W^{-1,2}(0,T;H)$ - but this is kinda handwaving and seems not correct.
So, how to get from $L^1(0,T;V')$ to $H^{-1}(0,T;H)$?
Right, here the definition of the spaces: $$H:=\{u \in L^2(\Omega)^d : \text{div} u=0, \gamma_\nu u =0 \}$$ $$V:=\{u \in H_0^1(\Omega)^d : \text{div} u=0 \}$$
EDIT: Posted my attempt as an answer for future reference if someone got a similar question.
See the comments for the hint of gerw. I did the following:
\begin{align} ||u'||_{H^{-1}(0,T;H)}&=\sup \frac{|\langle u', \phi \rangle_{H^{-1}(0,T;H),H_0^1(0,T;H)}|}{||\phi||_{H_0^1(0,T;H)}} \\ &=\sup \frac{|(u', \phi )_{L^2(0,T;H)}|}{||\phi||_{H_0^1(0,T;H)}} \\ &=\sup \frac{|(u, \phi' )_{L^2(0,T;H)}|}{||\phi||_{H_0^1(0,T;H)}} \\ &\leq\sup \frac{||u||_{L^2(0,T;H)} ||\phi||_{H_0^1(0,T;H)}}{||\phi||_{H_0^1(0,T;H)}} \\ &=||u||_{L^2(0,T;H)} \end{align}
where I used the Gelfand-Triple $$H_0^1(0,T;H) \hookrightarrow L^2(0,T;H) \hookrightarrow H^{-1}(0,T;H)$$