I'm trying to take the Fourier transform of this equation $f_t + \nabla \cdot (gf)=0$, because I want to know whether I can get kernel of this equation. Here we assume $\nabla \cdot g=0$. For example, we can use Fourier transform on heat equation $u_t - \Delta u =0$ to get $\hat u = e^{-|\xi|^2 t}u_{0}$.
So far, I can get $\hat f_t = \xi \hat{gf}(\xi)$ but don't know how to proceed to get $\hat f$.
We assume $g$ is a vector function in $\mathbb R^2$ and $f$ is a scalar function in $\mathbb R^2$.
Update: here I meant $f(x,t),g(x,t) \in \mathbb R^2 \times (0, \infty)$.