Assume that a measurable vector field $D$ of $\mathbf R^3$ satisfy $\nabla\cdot D = 0$ in the weak sense, that is, $$\int D \cdot \nabla \varphi = 0$$ for every test function $\varphi$. Does this imply that if $\Omega$ is a bounded domain with smooth boundary, we'll have $$\int_{\partial \Omega} D \cdot {\bf n} \ ds = 0?$$
Note: If $D$ is $C^1$, then this holds true because $\int D\cdot \nabla \varphi = \int (\nabla \cdot D) \ \varphi = 0$ for every test function $\varphi$ implies $\nabla\cdot D = 0$.