An ordinary differential equation that generalizes the notion of "path of steepest descent." For questions on "gradients" of a function, use (multivariable-calculus) instead.
Let $H$ be a Hilbert manifold. A gradient flow on $H$ is the ordinary differential equation for $u: I \to H$, where $I\subseteq \mathbb{R}$ is an interval, given by $$ \frac{\mathrm{d}}{\mathrm{d}t} u = -(\nabla F)(u) $$ where $F:H \to \mathbb{R}$ represents the potential or height.
The motivation is that of steepest descent in Euclidean spaces, and is connected with morse-theory. Imagine $F$ is some real valued function defined over $\mathbb{R}^2$, the vector $-\nabla F$ is a vector field that points in the direction of the steepest descent if we look at the graph of $F$ as describing a field of heights over $\mathbb{R}^2$. The gradient flow is the generalization of this to higher (possibly infinite) dimensions.
Certain partial differential equations can be described as a gradient flow in infinite dimensions. For example, consider $H$ being the Sobolev space $H^2(\mathbb{R}^d)$ of functions whose derivatives up to second order belong to $L^2$. If we define the function $F(u) = \int_{\mathbb{R}^d} |\nabla u|^2 ~\mathrm{d}x$, a formal computation shows that its gradient (relative to the $L^2$ inner product) is
$$ (\nabla F)(u) = \triangle u $$
where $\triangle$ is the usual Laplacian. That is to say, the gradient flow for the "energy functional" $F(u)$ is in fact the linear heat equation on $\mathbb{R}^d$.
