If $u \in L^2(0,T;X_0)$ with $u_t \in L^2(0,T;X_2)$, then is $u \in L^\infty(0,T;X_1)$?

Question

Let $X_0 \subset X_1 \subset X_2$ be continuous embeddings, with $X_0 \subset X_1$ compact.

Suppose $u \in L^2(0,T;X_0)$ with $u_t \in L^2(0,T;X_2)$.

Is then $u \in L^\infty(0,T;X_1)$?

To apply Aubin-Lions, one needs $u \in L^\infty(0,T;X_0)$ -- but this gives a compact embedding. I only need the weaker result that $u$ belongs to $L^\infty(0,T;X_1)$.

I tried many books with no luck.

Answer 1

This is true if $X_1=[X_2,X_0]_{1/2}$ (complex interpolation space), and this result is sharp. You will find this in many texts, e.g. Lions-Magenes.