When all electric charges are stationary (i.e. $\frac{\partial\rho}{\partial t}=0$ and $\mathbf{J}=0$), the Maxwell equations
$$\pmb{\nabla}\cdot\mathbf{E}=\rho/\varepsilon_0,$$ $$\pmb{\nabla}\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t},$$ $$\pmb{\nabla}\cdot\mathbf{B}=0,$$ $$\pmb{\nabla}\times\mathbf{B}=\mu_0\mathbf{J}+\varepsilon_0\mu_0\frac{\partial\mathbf{E}}{\partial t},$$ are directly reduced to
$$\pmb{\nabla}\cdot\mathbf{E}=\rho/\varepsilon_0,$$ $$\pmb{\nabla}\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t},$$ $$\pmb{\nabla}\cdot\mathbf{B}=0,$$ $$\pmb{\nabla}\times\mathbf{B}=\varepsilon_0\mu_0\frac{\partial\mathbf{E}}{\partial t},$$
where $\mu_0$ and $\varepsilon_0$ are physical constants. But from physics we know that when the charges are stationary, the Maxwell equations can be simplified further, to the following equations of electrostatics:
$$\pmb{\nabla}\cdot\mathbf{E}=\rho/\varepsilon_0,$$ $$\pmb{\nabla}\times\mathbf{E}=0,$$ $$\pmb{\nabla}\cdot\mathbf{B}=0,$$ $$\pmb{\nabla}\times\mathbf{B}=0.$$
Now the problem I'm thinking about is this: Let's forget all we known from physics, and consider everything from the perspective of pure mathematics. We have the first set of equations and we are given the conditions $\frac{\partial\rho}{\partial t}=0$ and $\mathbf{J}=0$. How can we derive the last set of equations?