I am reading Navier-Stokes Equations by Roger Temam and there is some step in a proof I don't understand. Let me recall the settings.
Let $X, X_0, X_1$ three Banach spaces with $X_0, X_1$ reflexive, such that $$X_0 \subset X \subset X_1$$ continuously and the inclusion $X_0 \subset X$ is compact. Let $T > 0$ a fixed finite number and $\alpha_0, \alpha_1 > 1$. Let us define $$\mathcal Y = \left\{v \in L^{\alpha_0}(0,T; X_0)~\big|~ v' \in L^{\alpha_1}(0,T;X_1)\right\}$$ endowed with the norm $$\|v\|_{\mathcal Y} = \|v\|_{L^{\alpha_0}(0,T; X_0)} + \|v'\|_{L^{\alpha_1}(0,T;X_1)}$$ which makes it a Banach space.
I am reading the proof that $\mathcal Y \subset L^{\alpha_0}(0,T; X)$ is compact. At some point in the proof, the author uses the fact that $\mathcal Y \subset C([0,T], X_1)$ continuously and I do not understand why. I know the following lemma:
Lemma. Let $u, v \in L^1(0,T; X)$, for $X$ a Banach space, such that $$\int_0^T u(t) \phi'(t)dt =-\int_0^T v(t) \phi(t)dt,$$ for all $\phi \in C^\infty_0([0,T])$. Then $u$ is a.e. equal to a continuous function from $[0, T] $ into $X$.
Therefore the inclusion $$\mathcal Y \subset C([0,T], X_1)$$ is trivial by the lemma. However, I do not see how could I show that it is continuous. Indeed, to this end, we should be able to show that there exists a constant $C>0$ such that, for all $u \in \mathcal Y$, $$\|v\|_{C([0,T], X_1)} \le C \|v\|_{\mathcal Y},$$ which means $$\|v(t)\|_{X_1} \le C(\|v\|_{L^{\alpha_0}(0,T; X_0)} + \|v'\|_{L^{\alpha_1}(0,T;X_1)}), \quad \forall t \in [0,T].$$ I tried many things but I did not find anything.. Any help ?