Deducing $L^4$ convergence in the space-time-cylinder

Question

I have a $f_m\in L^2(0,T;H^4(\Omega))\cap W_{\frac{4}{3}}^1(0,T;L^2(\Omega))$ where $\Omega\subset \mathbb{R}^n$ bounded.

I want to show that

$$\nabla f_m\to \nabla f \text{ as }m\to\infty\qquad\text{ in }L^4((0,T)\times \Omega).$$

I also know that $$f_m\to f \quad \text{ in }L^2(0,T;H^3(\Omega))$$ and $$f_m \text{ is bounded in }L^\infty(0,T;H^2(\Omega))\cap L^2(0,T;H^4(\Omega).$$

I am geniunely confused because this looks obvious to me with all the assumptions present, yet I am having trouble to make it into a rigorous argument (which really bugs me since I should be able to do this since my introduction to analysis course...).

Can I deduce this by using a weak convergence argument of some kind? Then I'm having trouble moving from $L^p(0,T;H^q(\Omega)$ to $L^4((0,T)\times\Omega)$ Or is this just writing down the $L^4$-norm and estimating?

Any hints would be appreciated.

Answer 1

You have convergence in $L^2(0,T; H^2(\Omega))$ and boundedness in $L^\infty(0,T; H^2(\Omega))$. Hence you can use Hölder's inequality to obtain convergence in $L^4(0,T; H^2(\Omega))$. If the space dimension is not too high, this embeds into $L^4(0,T; L^4(\Omega))$.