Is $\int_{t_1}^{t_2} \iiint_V \iiint_V \left(\vec{J}\left(\vec{r}_1,t\right) \cdot \vec{J}_t\left(\vec{r}_2,t\right)\right) $ zero if... In a finite volume $V \in \mathbb{R}^3$, there is a field  $\vec{J}\left(\vec{r},t\right)$. On the surface $\partial V$, $\vec{J}\left(\vec{r},t\right) = 0$.
The variable $\vec{r}$ represents position, and $t$ is, of course, time. Now say there are two time instants $t_1$ and $t_2$ such that for all $\vec{r} \in V$, $$\vec{J}\left(\vec{r},t_1\right) =\vec{J}\left(\vec{r},t_2\right)$$
Consider the following integral:
$$
   \int_{t_1}^{t_2} 
   \iiint_V
   \iiint_V
     \left[
      \vec{J}\left(\vec{r}_1,t\right) 
      \cdot 
      \frac {\partial \vec{J}\left(\vec{r}_2,t\right)} {\partial t}
     \right]
   \space dV\left(\vec{r}_1\right)
   \space dV\left(\vec{r}_2\right)
   \space dt
$$
Does this integral always evaluate to zero? Any proof?
Context: comes from classical electrodynamics, $\vec{J}$ is current density.
 A: Just so this question doesn't stay 'unanswered', here's consolidating the answer(s) given by @achillehui (all credit for the answer go to him)
First, define:
$$\vec{Y}\left(t\right)=
   \iiint_V 
      \vec{J}\left(\vec{r},t\right) 
   \space dV\left(\vec{r}\right)
$$
So $\vec{J}\left(\vec{r},t_1\right) = \vec{J}\left(\vec{r},t_2\right)$
for all $\vec{r}\in V$
implies $\vec{Y}\left(t_1\right) = \vec{Y}\left(t_2\right)$ also.
By Fubini's theorem, the integral in the question becomes:
$$
\int_{t_1}^{t_2}
   \left[ 
    \left(
     \iiint_V
      \vec{J}\left(\vec{r},t\right) 
     \space dV\left(\vec{r}\right)
    \right)
   \cdot
    \left(
     \iiint_V
      \frac {\partial \vec{J}\left(\vec{r},t\right)} {\partial t}
     \space dV\left(\vec{r}\right)
    \right)
   \right]
   \space dt
$$
Which becomes:
$$
\int_{t_1}^{t_2}
   \left[ 
    \left(
     \iiint_V
      \vec{J}\left(\vec{r},t\right) 
     \space dV\left(\vec{r}\right)
    \right)
   \cdot
    \left(
     \frac {d} {dt}
     \iiint_V
      \vec{J}\left(\vec{r},t\right)
     \space dV\left(\vec{r}\right)
    \right)
   \right]
   \space dt
$$
Which is:
$$
\int_{t_1}^{t_2}
   \left[ 
    \vec{Y}\left(t\right)
   \cdot
    \left(
     \frac {d} {dt}
     \vec{Y}\left(t\right)
    \right)
   \right]
   \space dt
$$
Which is:
$$
\frac{1}{2} \left[|\vec{Y}(t)|^2\right]_{t1}^{t2}
$$
Which is zero because $\vec{Y}\left(t_1\right) = \vec{Y}\left(t_2\right)$
