Let $V=H^1(\Omega)$ and $U = L^{\infty}(\Omega)^2$. Take for example, a differential operator, $L:U\times V \rightarrow V^* $ such that:
\begin{equation} L(u,y) = \Delta y + \vec{u}(x)\nabla y \end{equation}
How does one go about defining the derivative of such a operator with respect to the the parameter $u$ or $y$. Like in the case of finding the adjoint of a differential operator do we need to specify the domain of the operator in terms of the boundary conditions? Does anything change if the $u$ and the boundary conditions are also time dependent?