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Am I right that the gradient flow of a functional $E$ is $$f_t = -\nabla E(f).$$ Solving this for $f$ gives you a minimiser of $E$ in some way?

Here the $\nabla$ denotes the gradient or the first variation or Gateaux derivative or whatever is appropriate.

What is meant by "$L^2$ gradient (flow)" or "$H^{-1}$ gradient (flow)?"


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up vote 3 down vote accepted

If $E$ is a Frechet differentiable functional on a Hilbert space, the gradient of $E$ at $u$ is the element $w$ of the Hilbert space satisfying $E'(u)v = (w,v)$ for all $v$ in the Hilbert space. The $E$ gradient flow starting of $u$ is the solution $\eta(t)$ of the diffential equation $\frac{d}{dt}I(\eta) = -\nabla E(\eta), \eta(0) = u$. Sorry, I don't know how to make the accent mark in Frechet, nor do I know how this works more generally in Banach spaces. If $\eta(t)$ converges to some $w$ as $t\to \infty$, then that $w$ is a critical point of $E$ (not necessarily a minimizer). If someone says ''$L^2$ gradient flow'' or ''$H^{-1}$ gradient flow" they mean that the Hilbert space is $L^2$ or $H^{-1}$, but the gradient flow depends on the functional.

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