# A general solution to continuity equation

Let us write the standard continuity equation $$\frac{\partial \rho}{\partial t} + \vec{\nabla} \cdot \vec{\jmath} = 0.$$

Should the relation $\vec{\jmath} = \rho \vec{v}$ be considered as a general solution of the continuity equation? If so, how to derive it?

• $\vec j=\rho \vec v$ is not a solution of the continuity equation, it is a parametrization of the variables present in it. With just this continuity equation, you can't get any solution because you have 1 (scalar) equation and 4 indepent variables. – Hydro Guy May 4 '14 at 19:44
• Generally one numerically solves this problem. If you have strict boundary conditions and initial conditions, it may be possible to analytically solve for $\rho$. – Kyle Kanos May 4 '14 at 21:17
• I was thinking more along the lines of... The general solution of the wave equation in 1D is of the form $f(x \pm ct)$, with $f$ arbitrary. – user17116 May 4 '14 at 22:25