As I know, this equation is a consequence of the Coulomb law + Gauss's theorem in the ELECTROSTATICS. It is impossible to measure \rho directly especially if the charged body moves.

It is an intersting question. In electrostatics Div(D) = free charge density comes from Coulomb's force law. However, it is important to know that Faraday did an experiment by enclosing a charged spherical conductor in dielectric shells, demonstrating that divergence of the displacement vector to be independent of the dielectric. One can find this experiment described in old text books on electricity and magnetism. 

It is not correct to state that this divergence law can not be inferred  from electrodynamics. As a matter of fact it can be inferred directly  from curl (H) = J + dD/dt = Conduction current density + displament current density. Diverdence of the equation yields the current continuity equation, which is the law of conservation of charge. For the the conservation of charge div(D) must be the free charge density. This realization by Maxwell led him to add the displament current to the Ampere's law in magnetostatics, giving us the famous Maxell's equations for electrodynamics.

But Faraday's experiment belongs rather to ELECTROSTATICS (magnetostatics) than to the electrodynamics. The author asks about experimental fact from the electrodynamics.

Dear Alberto Díaz

Thank you for your descriptions. I am interested in to know whether the physicist prior to Maxwell recognized this equation valid in electrodynamics as in electrostatics? If yes, on what basis? If no how did Maxwell extend the law for electrostatics to electrodynamics?

Dear All,

    As I pointed out earlier, Maxwell corrected Ampere's law for magnetostatics by adding the displacement current density. This led to make Ampere's law to conform with the current continuity relation (charge conservation). The charge coservation by the electrodynamic equations is the best confirmation of the div(D) = free charge density. A confirmation of modified Ampere's law is a confirmation of div(D) = free charge density. Measuring surface charge densities on conductors in transmission lines and waveguides and the associated electric fields could verify the law.

As I know, except Maxwellian approach there is the approach of Gauss--Riemann-Lorenz. In this approach the equation div D = \rho isn't needed. Curoiously but all write the Maxwell equations as a starting point and then use the approach of Gauss--Riemann-Lorenz to obtain expressions for the EM fields.

Dear Singh,

But, Maxwell's correction to Ampere's law was so that to be consistent with charge conservation.

If we consider  first Maxwell's equation in another form, then Maxwell correction should be another expression instead of ∂D/∂t.

Happy New Year

Dear Mohammad,

    As I had pointed out earlier, the charge conservation from electrodynamic equations follows only after adding the displacement current in the Ampere's law, a dicovery by Maxwell.

   Which other form you are referring to? The differential and integral forms are the ways of writing Maxwell equations. The both forms are equivalent; in the integral form the time derivative of the total displacement current threading an appropriate closed surface enters the equation while in the differential form it is the time derivative of the displacement current density at a point. The Kirchoff's current law represents the integral form. Thus the Kirchoff's current law applied to any time-varying circuit containing charge-storage devices (like a capacitor) could be an experimantal verification of the modified Ampere's law.

Happy new year