Let $u:\mathbb{R}^2\rightarrow\mathbb{R}^2$ be defined such that
$\begin{align*} \qquad\qquad\qquad\qquad\qquad\qquad\qquad u(x,y)=(u_1(x,y),u_2(x,y)) \end{align*}$.
Consider the operator
$\begin{align*} \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad A = (-\Delta)^{-1}(u\cdot\nabla) . \end{align*}$
How do I make sense of $A^0$? Is it simply the identity?
In general if we have an identity element $I$ for an operation $*$ in a set $G$ we can define
$A^0=I$ ( it's more a matter of defition in the end)
With this we can define the composition recursively $$A^0=I$$ $$A^{n+1}=A^{n}*A $$
With that $A^1=A^0*A=I*A=A$, and so on