What is the name of this PDE $\partial_t u = F(u) - \mathcal{L}_{\mathcal{V}}u$?

Question

I've come across the following PDE $$ \partial_t u = F(u) - \mathcal{L}_{\mathcal{V}} u $$ where the solution $u : \mathbb{R}^3 \times \mathbb{R} \to \mathbb{R}^3$ is a vector field in space $\mathbb{R}^3$ and time $\mathbb{R}$, which evolves according to a nonlinear function $F :\mathbb{R}^3 \to \mathbb{R}^3$ and the Lie derivative $\mathcal{L}_{\mathcal{V}}$ defined with respect to a smooth vector field $\mathcal{V}$ on $\mathbb{R}^3$. Does anyone know what this type of PDE this is called?