I'm reading the paper of Seregin, Euscariaza and Sverak about the smoothness of weak solutions $(u,p)$ of the Navier Stokes equations when $u \in L^{\infty}([0,T],L^3(\mathbb{R}))$.
Here is the link.
I have one question about the first step for the proof of Theorem 1.3. They claim that since $u$ is $L^2$ weakly continuous and belongs to $ L^\infty{[0,T],L^3(\mathbb{R})}$ we can deduce that for each time we have $u(t,\cdot)\in L^3(\mathbb{R})$, it is not so clear.
I read the book of Robinson, and he explained using duality, but it is not clear either.
Could you help me understanding this claim?
Thanks.
My idea:
I assume that since is weakly continous in $L^2$, I can approximate any fonction in $L^\frac32$ by something in $L^\frac32\cap L^2$ and therefore the funcion is weakly continous in $L^\frac 32$. This is enough for concluding that each time belongs to $L^3$