Various definitions of weak solutions and continuity property

Question

For example, consider the transport equation $$ \partial_tu+b\cdot\partial_xu=0, u(0,x)=u_0(x), $$ where $t\ge0,x\in\mathbb{R}^3$, $b\in W^{1,1}_{\rm{loc}}([0,\infty)\times\mathbb{R}^3)$.

It seems that the definitions of weak solutions depend on the forms of test function spaces. I wonder the difference and relationship of the following cases. More precisely, I want to figure it out the difference of the continuity at $t=0$ of them.

1. The space of test functions is defined as $C_c^{\infty}([0,\infty)\times\mathbb{R}^3)$.

2. The space of test functions is defined as $C_c^{\infty}((0,\infty)\times\mathbb{R}^3)$and the weak solutions are equipped with the continuity property in a sense at $t=0$, e.g., weakly in $L^2(\mathbb{R}^3)$.