Solving the continuity equation with divergence and time derivative

[lukaszh]

To solve
$$\frac{\partial\varrho}{\partial t}+\mathrm{div}(\varrho\vec{v})=0$$

 

[isabelle]

That is the continuity equation, it expresses the fact that the total time derivative of rho is zero. But if v is also an unknown vector function, then you have four unknowns and one equation so the system is underdetermined.