It is standard in many functional analysis, PDEs, and optimization contexts to consider the gradient of functionals $F(u)$ on some Hilbert space $H$ with with respect to its inner product $\langle\cdot,\cdot\rangle_H$, particularly when $H = L^2(\Omega)$, where the functional derivatives are easy to compute and gradients are easy to identify. For example, it is easy to verify that for $F(u) = \frac{1}{2}\int_\Omega |\nabla u|^2 dx$, we have $\nabla F(u) = -\Delta u$, and for $F(u) = \frac{1}{2}\int_\Omega (u - u_d)^2 dx$, we have $\nabla F(u) = u - u_d$. Gradients and gradient flows have been considered in more exotic spaces, for instance, one can show that the gradient flow of entropy of a probability distribution with respect to the Wasserstein metric yields a heat equation.
My question is, is there any simple or canonical way to characterize the gradients of common functionals (such as those previouly above) when $H$ is a Sobolev Hilbert space $H^k(\Omega) = W^{k,2}(\Omega)$? When finding gradients in $L^2(\Omega)$ once can usually find gradients by inspection and liberal use of integration by parts, but identifying gradients in Sobolev spaces seems more involved for any examples that are not written precisely involving the inner product itself. Note that I am not specifying the exact form of the inner product (sum of derivatives, Fourier, etc.) as I am curious as to whether using a different equivalent inner product may yield easier characterization of gradients.
If this cannot be done, are there any results characterizing their gradients/gradient flows in contrast to those on $L^2(\Omega)$ or other function spaces?