What is the meaning of the space $L^\infty(0,T;L^2(\mathbb{R}))$.
In the article, it says $\sqrt{\rho}(u-\bar{u})\in L^\infty(0,T;L^2(\mathbb{R}))$. Does it mean that $\sup\limits_{t\in [0,T]}\left\lVert \sqrt{\rho}(u-\bar{u}) \right\rVert_{L^2(\mathbb{R})}\leq C$ and $C$ is independent of $T$
If $C$ is dependent of $T$, why in the article a integral like follows $$\sup_{t\in[0,T]}\int_\mathbb{R}\cdots+\int_0^T\int_\mathbb{R}\cdots$$ is controled by $C\varepsilon\ln(1+T)+C\varepsilon(1+\sup\limits_{t\in [0,T]}\left\lVert \sqrt{\rho}(u-\bar{u}) \right\rVert_{L^2(\mathbb{R})})+C \sup\limits_{t\in [0,T]}\left\lVert \sqrt{\rho}(u-\bar{u}) \right\rVert_{L^2(\mathbb{R})}$
and why the article says if $\sqrt{\varepsilon}\ln(1+T)\leq 1$ then the integral is less than $C$, and $C$ is independent of $\varepsilon,T$
the doi number of the article is 10.1080/03605302.2010.516785
or go to the link https://www.tandfonline.com/doi/abs/10.1080/03605302.2010.516785
My question arise from lemma 3.1