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?

  • 4
    $\begingroup$ $\vec j=\rho \vec v$ 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. $\endgroup$ – Hydro Guy May 4 '14 at 19:44
  • $\begingroup$ Generally one numerically solves this problem. If you have strict boundary conditions and initial conditions, it may be possible to analytically solve for $\rho$. $\endgroup$ – Kyle Kanos May 4 '14 at 21:17
  • $\begingroup$ 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. $\endgroup$ – user17116 May 4 '14 at 22:25

Solutions of the continuity equation are discussed in a paper C.F.Clement in Series A, Mathematical and Physical Sciences, Vol. 364, No. 1716 (Dec. 12, 1978), pp. 107-119 (link to pdf)

  • $\begingroup$ Maybe you could discuss some of the relevant results? Links aren't really fit as answers. $\endgroup$ – anon01 Jun 21 '16 at 21:08

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