In Villani's Hypocoercivity, there is a statement as follows,
... even if $\nabla V$ is only continuous: the differential operator $\nabla V(x)\cdot \nabla_v$ makes distributional sense
I don't understand how $\nabla V(x)\cdot \nabla_v$ makes distributional sense even though $\nabla V$ is only continuous. Shouldn't $\nabla V(x) \in C^\infty$ if it would make distributional sense? Any help or any reference would be very much appreciated!