Relation between bochner space $L^1(I,X)$ and $C(I,X)$

Question

I am new in Bochner spaces and I have the following problem. I am not able to prove, that if $u \in L^1((0,1),X),u' \in L^1((0,1),X) $ then $u \in C([0,1],X)$, where $X$ is Banach space and $L^1((0,1),X) $ Bochner space.

Answer 1

You are mostly asking about the embedding $$W^{1,1}(I;X) \hookrightarrow C(I;X),$$ which is quite standard and can be seen in most books that deal with Bochner spaces and PDES, see e.g. Roubicek or Cazenave&Haraux.

Let $I=[a,b]$ be some interval and let $u \in W^{1,1}(I;X)$, then $u' \in L^1(I;X)$ and we set $v(t):=\int_a^t u'(s) \, \text{d}s$. Then

$$\| v(t_2)-v(t_1)\|_X = \left\| \int_{t_1}^{t_2} u'(s) \, \text{d}s \right\|_X \leq \int_a^b \|u'(s)\|_X \, \text{d}s$$

for all $a \leq t_1 \leq t_2 \leq b$, showing $t \mapsto v(t):I \to X$ is continuous. Since $u'=v'$, we have $v=u+c$, $c \in X$, and therefore $u$ is continuous, too.

Now, you can simply prove the continuity of the embedding, i.e. $\|u\|_{C(I;X)} \leq C \|u\|_{W^{1,1}(I;X)}$, you can look at the given references.