I have answers for this following first order PDE this one $$\left\{\begin{array}{lll} f_{t}+xf_{y}=0\\ f|_{t=0}=f_{0}(x,y) \end{array}\right.$$
But I just wonder what is the geometric meaning for this PDE .
Any comment or suggestion will be appreciated .Thanks for considering my request .
This looks like a two dimensional balance law or here better conservation law. With $\boldsymbol{x}=(\boldsymbol{x}_{1},\boldsymbol{x}_{2}):=(x,y)$ define $\vec{v}(t,\boldsymbol{x})=\begin{pmatrix} 0\\\boldsymbol{x}_{1}\end{pmatrix} $, the initial value problem can be written as \begin{align*} f_{t}(t,\boldsymbol{x})+ \operatorname{div}\left(f(t,\boldsymbol{x})\vec{v}(t,\boldsymbol{x})\right)&=0\\ f(t,\boldsymbol{x})&=f_{0}(\boldsymbol{x}) \end{align*} Thus, the initial datum $f_{0}$ is conserved and moves in $\vec{v}$ direction dependent on $\boldsymbol{x}_{1}$ spatial coordinate.