I often hear people say that the solution to the continuity equation is a current or can be viewed as a current but I never understood how exactly one can rewrite a continuity equation to make use of one such formulation.
Given a vector field $$ b:\mathbb{R}^{n+1}\to\mathbb{R}^n $$ We want to find curves of measure that solves the distributional equation $$ \frac{d}{dt} \mu_t+\mathrm{div}(b_t\mu_t). $$ I do understand that one can abstractly say that the above is a requirement on the boundary of a 1-current over space-time, i.e. a current $$T:\Omega_c(\mathbb{R}^n\times\mathbb{R})\to\mathbb{R}$$ Which is represented by $$T=\int_\mathbb{R} (1\ \ b_t)\mu_t$$ And satisfies $$\partial T=0.$$
What I do not understand is how one can make use of this formulation and obtain solutions or even if this is the formulation people speak about.
Even just a reference where I can look up this kind of presentation would be nice.