Standard Interpolation between Bochner spaces

Question

I've read the following in a few papers:

Given: Let $\Omega \subset R^d$ be a Lipschitz domain. A sequence $f_n$ converges strongly to $f$ in $L^2(0,T;L^2(\Omega))$ and weakly in $L^2(0,T;H^2(\Omega))$.

Then by a standard interpolation argument it follows that $f_n$ converges strongly to $f$ in $L^2(0,T;H^1(\Omega))$.

Now my question is what is meant by standard interpolation argument?

Thanks for your answers!

Answer 1

You may want to have a look for a book on interpolation theory, there is, e.g., a great introduction by Tartar.

There are, typically, two interpolation results: $$\|u\|_{L^p} \le \|u\|_{L^r}^\mu \, \|u\|_{L^s}^{1-\mu},$$ (Lebesgue scale), if $1/p = \mu/r + (1-\mu)/s$ and $$\|u\|_{H^1} \le C \, \|u\|_{L^2}^{1/2} \, \|u\|_{H^2}^{1/2}$$ (Sobolev scale).

Your proof requires the second one (use boundedness in $H^2$).