What is a gradient flow?

November 5, 2010

What is a gradient flow?

Filed under: Uncategorized — Tags: flow — Ngô Quốc Anh @ 18:40

In this entry, we shall discuss the following question: “What is a gradient flow?”.

Let $(H, \langle {\cdot, \cdot}\rangle)$ be a Hilbert space with norm $\| \cdot\|$ . Let $\phi$ be a convex, lower semi-continuous functional defined on a dense domain $\text{Dom}(\phi) \subset H$ .

Definition. The subdifferential of $\phi$ at a point $u \in \text{Dom}(\phi)$ is the set $\partial \phi(u) \subset H$ defined by

$\partial \phi (u) =\{p \in H: \phi(v) \geqslant \phi(u)+\langle p, v-u\rangle \quad \forall v \in \text{Dom}(\phi)\}$ .

Vector $p$ is called subgradient at $u$ , thus, the set of all subgradients at $u$ is called the subdifferential at $u$ .

The geometric meaning of subdifferential is as the set of all possible “slopes” of affine hyperplanes touching the graph of $\phi$ from below at the point $u$ . Thus, $\phi$ is differentiable at $u$ iff its subdifferential at $u$ contains exactly one vector as its derivative at that point.

We now recall the classical definition of gradient flow on a Hilbert space.

Definition. A function $u\in AC_{loc}((0,+\infty); H)$ , the class of absolutely continuous from $(0,+\infty)$ to $H$ , is a gradient flow of the convex, lower semi-continuous functional $\phi$ iff the differential inclusion

$u_t \in -\partial \phi(u(t))$

is satisfied almost everywhere with respect to $t$ .

In practice for a given flow

$u_t = F(u)$ ,

if $\phi$ is differentiable and is given as an integral over some domain, say, $\Omega$ , we simply verify there is some number $\lambda$ so that

$\displaystyle \int_\Omega u_t F(u) = \lambda \frac{d}{dt}\phi(u(t))$ .

If $\lambda >0$ , the flow is called positive gradient flow. Now we discuss some examples.

Example 1. The following semilinear heat equation

$u_t =\Delta u+|u|^{2^\star-1}, \quad x \in \Omega$

corresponds formally to the $L^2$ gradient flow associated to the energy functional

$\displaystyle E(u)=\frac{1}{2}\int_\Omega |\nabla u|^2dx-\frac{1}{2^\star}\int_\Omega |u|^{2^\star}dx$ .

Here we assume the boundary and initial conditions are all zero just for simplicity.

Observe that

$\displaystyle\frac{1}{2}\frac{d}{{dt}}\int_\Omega {|\nabla u{|^2}dx} = \int_\Omega {\nabla u \cdot \nabla {u_t}dx} = - \int_\Omega {{u_t}\Delta udx}$

and

$\displaystyle\frac{1}{{{2^ \star }}}\frac{d}{{dt}}\int_\Omega | u{|^{{2^ \star }}}dx = \int_\Omega | u{|^{{2^ \star } - 1}}{u_t}dx$ .

Therefore

$\displaystyle\frac{d}{{dt}}E(u) = - \int_\Omega {{u_t}\left[ {\Delta u + |u{|^{{2^ \star } - 1}}} \right]dx}$

i.e.

$\displaystyle u_t =\Delta u+u^{2^\star-1}$

is the $L^2$ gradient flow associated to the energy functional

$\displaystyle E(u)=\frac{1}{2}\int_\Omega |\nabla u|^2dx-\frac{1}{2^\star}\int_\Omega |u|^{2^\star}dx$ .

Example 2. Heat flow for Nirenberg’s problem

$u_t = \alpha f - K$

is also a gradient flow for the following functional

$\displaystyle E_f(u)=\int_{\mathbb S^2} (|\nabla u|^2+2u)dv_g - \log\left(\int_{\mathbb S^2} fe^{2u}dv_g\right)$ .

For the details of this flow, we refer the reader to a paper due to Michael Struwe published in Duke Math. J. in 2005 [here].

Example 3. $Q$ -curvature flow

$\displaystyle \frac{\partial}{\partial t}g=-\left(Q-\frac{\overline Q}{\overline f}f\right)$

is a gradient flow for some functional. For details, we refer the reader to a paper due to Simon Brendle published in Ann. of Math. in 2003 [here].

Source: Steepest descent flows and applications to spaces of probability measures by Luigi Ambrosio.

Comments (1)

1 Comment »

[…] E[u]=int w(u)+epsilon ^2 |nabla u|^2 ) $$ I tried to follow the definition of gradient flow from : https://anhngq.wordpress.com/2010/11/05/what-is-a-gradient-flow/ but I got stucked and […]

Pingback by gradient flow -cahn hilliard - MathHub — May 14, 2016 @ 17:50

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Ngô Quốc Anh

November 5, 2010