Suppose first two Maxwell equations (divergence E, B) only valid at one instance, how other two (curl E, B) ensure they are valid all the time? If $\nabla · \mathbf E = \displaystyle\frac{\rho}{\epsilon_0}$ and $\nabla · \mathbf B = 0$ is valid only at $t = t_0$.
How to use the remaining two Maxwell's equations:
\begin{align}
\nabla \times \mathbf E &= - \frac{\partial \mathbf B }{\partial t}\\
\nabla \times \mathbf B &= \mu_0 \mathbf J +\mu_0 \epsilon_0 \frac{\partial\mathbf E }{\partial t}
\end{align}
to ensure they are valid at all times?
Thank you!
 A: This is a question in the initial-value formulation of fields.  Suppose we specify the fields $\vec{E}$ and $\vec{B}$ on some initial-data slice of spacetime ($t = t_0$).  Suppose also that $\vec{E}$ and $\vec{B}$ satisfy the constraints
$$
\vec{\nabla} \cdot \vec{E} - \frac{\rho}{\epsilon_0} = 0\qquad \vec{\nabla} \cdot \vec{B} = 0
$$
at $t = t_0$.  These two equations should be thought of as constraint equations, rather than evolution equations, because they do not contain any time derivatives of the fields;  if you know $\rho$ on the initial data surface, then you can "check" whether $\vec{E}$ and $\vec{B}$ satisfy these conditions without knowing the values of the fields at any time other that $t = t_0$.
The question being asked is whether the "evolution" equations
$$
\vec{\nabla} \times \vec{E} + \frac{\partial \vec{B}}{\partial t} = 0, \qquad \vec{\nabla} \times \vec{B} -\frac{1}{c^2} \frac{\partial \vec{E}}{\partial t} = \mu_0 \vec{J}
$$ 
preserve these constraints.  In other words, if the fields satisfy the constraint at $t_0$, and the fields evolve according to the evolution equations, do they still satisfy the constraint at $t = t_0$?
To solve this, you need to use the evolution equations to prove that
$$
\frac{\partial}{\partial t} \left[ \vec{\nabla} \cdot \vec{E} - \frac{\rho}{\epsilon_0} \right] = 0\qquad \frac{\partial}{\partial t} \left[ \vec{\nabla} \cdot \vec{B} \right] = 0
$$
at all times.  If this is true, then we can conclude that establishing the constraint equations at some initial time is sufficient to show that they hold for all times.  (The quantities in brackets are zero at $t = t_0$, and their time derivative is zero at all times, therefore these quantities must be zero at all times.)
The actual proof is actually pretty straightforward once you know what you're aiming for, and this is a standard exercise in mathematical physics, so I'll omit the proof for now.  I will say, though, that you will need to assume the continuity equation holds at all times, and that time derivatives commute with spatial derivatives.
