Let $\Omega$ be an open connected convex subset of $\mathbb R^d$. Let $\mathcal P (\Omega)$ be the space of Borel probability measures on $\Omega$. Let $C_0 (\Omega)$ be the space of real-valued continuous functions on $\Omega$ that vanish at infinity. We endow $\mathcal P (\Omega)$ with the topology of weak$^*$ convergence, i.e., $\mu_n \to \mu$ if and only if $\int \varphi \, \mathrm d \mu_n \to \int \varphi \, \mathrm d \mu$ for every $\varphi \in C_0 (\Omega)$.

Let $\mu : [0, T] \to \mathcal P (\Omega), t \mapsto \mu_t$ be continuous. We fix a Borel vector field $v:[0, 1] \times \Omega \to \mathbb R^d$ such that $v_t := v(t, \cdot) \in L^1 (\Omega, \mu_t, \mathbb R^d)$ for all $t \in [0, 1]$.

We consider the continuity equation $$ \partial_t \mu_t+\operatorname{div} (v_t \mu_t)=0.\label{a}\tag{$\ast$} $$

At page 123 of Santambrogio's book

*Optimal Transport for Applied Mathematicians*, the author defines the weak solution of \eqref{a} as $$ \begin{align} \int_0^1 \int_\Omega \partial_t \phi_t ( x) \, \mathrm d \mu_t (x) & + \int_0^1 \int_\Omega \nabla \phi_t (x) \cdot v_t (x) \, \mathrm d \mu_t (x) \, \mathrm d t = 0,\\ &\forall \phi \in C^\infty_c ((0, 1) \times \Omega). \end{align} \label{1}\tag{1} $$At page 3 of this note, the author defines the weak solution of \eqref{a} as $$ \begin{align} \frac{\mathrm d}{\mathrm d t} \int_\Omega \phi_t ( x) \, \mathrm d \mu_t (x) & + \int_\Omega \nabla \phi_t (x) \cdot v_t (x) \, \mathrm d \mu_t (x) = 0,\\ & \forall \phi \in C^\infty_c ((0, 1) \times \Omega). \end{align}\label{2}\tag{2} $$

At page $2$ of this note, the author defines the weak solution of \eqref{a} as $$ \begin{align} \frac{\mathrm d}{\mathrm d t} \int_\Omega \phi ( x) \, \mathrm d \mu_t (x) & + \int_\Omega \nabla \phi (x) \cdot v_t (x) \, \mathrm d \mu_t (x) = 0,\\ &\forall \phi \in C^\infty_c (\Omega). \end{align}\label{3}\tag{3} $$

The formulation \eqref{3} is somehow different from (\ref{1}, \ref{2}) because the test functions depend only on $x$.

Are the formulations (\ref{2}, \ref{3}) equivalent?

Thank you so much for your elaboration!

**Update** I have added below the screenshot of (\ref{2})