Why $t^{\frac{N}{2}\left(\frac{2}{N}-\frac{1}{q}\right)}N(\cdot) \in BC_w([0,\infty);L^q)$?

Question

In the article of "Kozono et. all (2016) pg. 11" with title: Existence and uniqueness theorem on mild solution to the Keller- Segel system coupled with the Navier Stokes fluid, the author can (under assumptions consting in the work, of course) the estimate $$\sup_{t>0}t^{\frac{N}{2}(\frac{2}{N}-\frac{1}{q})}\|N(t)\|_{L^q}\leq C\|n_0\|_{L_w^{\frac{N}{2}}}+C \sup_{\tau>0}\tau^{\frac{N}{2}(\frac{2}{N}-\frac{1}{q})}\|n(\tau)\|_{L^q}\times (1+\sup_{\tau>0}\tau^{\frac{N}{2}(\frac{1}{N}-\frac{1}{r})}\|\nabla c(\tau)\|_{L^r}+\sup_{\tau>0}\tau^{\frac{N}{2}(\frac{1}{N}-\frac{1}{r})}\|\nabla v(\tau)\|_{L^r}+\sup_{\tau>0}\tau^{\frac{N}{2}(\frac{1}{N}-\frac{1}{p})}\|u(\tau)\|_{L^p}), \ \forall t >0.$$ Seem obvious for the author that this estimate above implies that $t^{\frac{N}{2}\left(\frac{2}{N}-\frac{1}{q}\right)}N(\cdot) \in BC_w([0,\infty);L^q)$. I cant see cause this functions is weak-star continuous. Can somebody help me?