Let $\rho^k_t(x),\rho_t(x) ~: \mathbb{R}_+\times \mathbb{R}^d\to \mathbb{R}$ be some functions. I am reading an article that says proves two types of convergence of $\rho^k$ to $\rho$, but cant see the difference between them.
$\underline{\text{Statement 1 : }}$
For any $T>0$ we have that
\begin{equation} \rho^k \to_{k\to \infty} \rho ~~\text{weakly in } ~L^1((0, T)\times \mathbb{R}^d), \end{equation}
where $\rho$ is the solution ( to some equation ) with initial condition $\rho_0$.
$\underline{\text{Statement 2 : }}$
\begin{equation} \rho^k_t \to_{k\to \infty} \rho_t ~~\text{weakly in } ~L^1(\mathbb{R}^d)~ \text{for any} t>0, \end{equation}
and as $t\to 0$ $\rho_t\to \rho_0 $ in $L^1(\mathbb{R}^d)$.
Could anyone highlight the major differences in these two statements?