Imagine that we start with a linear functional $f$ which takes $X$ to $\mathbb{R}$. If we complete that to a space $\hat{X}$ which is obtained by equipping it with an inner product and taking the closure, then $f \in (\hat{X})^{*}$. (Recall that Hilbert spaces are defined to be complete).
The Riesz representation theorem for the connection between a Hilbert space and its continuous dual space then then says $f(x)=\langle \nabla f, x\rangle$ for any $x$ in $\hat{X}$ and any $\nabla f$ in $(\hat{X})^{*}$. Obviously, that inner product could be the $L^2$ norm or whichever you choose depending on the space. $f$ can always be written $\langle \nabla f, x \rangle$ and $f$ having gradient just means $\nabla f \in X$.
So a $L^2$ gradient flow is just a flow for a $L^2$ gradient, where the definition of a gradient flow is given by Stefan above. As an example from differential geometry, take a map $\pi$ from $E$ to a compact, oriented manifold $M$ where there is a bundle metric $\langle \cdot, \cdot\rangle_E$. In this case, the $L^2$ norm has to take $M$ to $\text{Sym}^2 (T^{*} M)$, so it takes the form
$\langle s, t \rangle_{L^{2}} = \int_M \langle s(x), t(x) \rangle_E \: \text{d}v(x)$
where $s,t \in \Gamma(E)$.