I'm reading Filippo Santambrogio's Optimal transport for Applied Mathematicians and I've come across the following (on page 123). The context is that $\mathbf{v}_t$ is a vector-valued velocity function and $\varrho_t$ is a density measure.
We say that a family of pairs of measures/vector fields $(\varrho_t, \mathbf{v}_t)$ with $\mathbf{v}_t \in L^1(\varrho_t; \mathbb{R}^d)$ and $\int_0^T ||\mathbf{v}_t||_{L^1(\varrho_t)} dt < +\infty$ solves the continuity equation $$ \partial_t \varrho_t \, + \, \nabla \cdot (\varrho_t\mathbf{v}_t) = 0 $$ on $(0,T)$ in the distributional sense if, for any bounded and Lipschitz test function $\phi \in C_c^1((0,T)\times \overline{\Omega})$, we have $$ \int_0^T \int_\Omega (\partial_t\phi) d\varrho_tdt + \int_0^T \int_\Omega \nabla \phi \cdot \mathbf{v}_td\varrho_t dt = 0$$ This formulation includes no-flux boundary conditions on $\delta \Omega$ for $\mathbf{v}_t$, i.e. $\varrho_t \mathbf{v}_t \cdot \mathbf{n} = 0$
My question is this: what does the boundary condition at the end ($\varrho_t \mathbf{v}_t \cdot \mathbf{n} = 0$) mean? It seems like we are multiplying a measure by a two vector fields. Does this give us a measure? Do we find the dot product of the vector fields first?
The zero scalar product $\mathbf{v}_t \cdot \mathbf{n} = 0$ would mean the velocity has no component normal to the boundary, so simply any flow at the boundary is precisely along the boundary, not across it.
With the density factor added, $\varrho_t \mathbf{v}_t \cdot \mathbf{n} = 0$ means the effective transport of mass accross the boundary is zero. That means you can have a non-zero cross-border velocity at some point or area, but the density there has to be zero. Generally, even iv $\mathbf{v}_t$ is not parallel to the boundary the mass flux across the border is zero.