What's a measure valued solution of a PDE?
For instance the Fokker-Planck equation \begin{align} \partial_t\mu_t+\sum_i\partial_i(b_i\mu_t)-\frac{1}{2}\sum_{ij}\partial_{ij}(a_{ij}\mu_t)=0 \end{align} it says that for a measure $\mu=\mu_(t,x)=\mu_t(x)$ being a solution of the above equation means \begin{align} \frac{d}{dt}\int_{\mathbb{R}^N}\phi(x)d\mu_t(x)=\int_{\mathbb{R}^N}\left(\sum_ib_i(t,x)\partial_i\phi(x)+\frac{1}{2}\sum_{ij}a_{ij}(t,x)\partial_{ij}\phi(x)\right)d\mu_t(x) \end{align} How do we get this? Also if compare this with the weak formulation, What's the connection between a measure valued solution and a distributional solution?
Many thanks!