3-variable transport equation:
\begin{align*} u_t + a u_x + b u_y =0 \qquad \text{on} \, [-3,3]^2 \times [0,+\infty) \end{align*}
where $a = -y(1-x^2-y^2)$ and $b = x(1-x^2-y^2)$ for $x^2+y^2 < 1$, otherwise $a = 0$ and $b = 0$. Thus the solution is time-independent on the boundary.
Initial condition:
\begin{align*} u(x,y,0) = u_0(x,y) = \sqrt{(x-1)^2 + (y-1)^2} - 1 \qquad \text{on} \, [-3,3]^2 \end{align*}
To solve this PDE I think the method of characteristics should be applied. But I don't know the particular implementation for the 3-variable case. Could you please show me how to find the analytical solution of this PDE problem step by step? (This is NOT homework)
Just like in the 2-dimensional case you get $$ \frac{dt}{ds}=1\\ \frac{dx}{ds}=a\\ \frac{dy}{ds}=b $$ which especially implies $\frac{d}{ds}(x^2+y^2)=0$ so that the factor $(1-x^2-y^2)$ is constant along the characteristics.