In this entry, we shall discuss the following question: “What is a gradient flow?”.
Let be a Hilbert space with norm . Let be a convex, lower semi-continuous functional defined on a dense domain .
Definition. The subdifferential of at a point is the set defined by
Vector is called subgradient at , thus, the set of all subgradients at is called the subdifferential at .
The geometric meaning of subdifferential is as the set of all possible “slopes” of affine hyperplanes touching the graph of from below at the point . Thus, is differentiable at iff its subdifferential at 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 , the class of absolutely continuous from to , is a gradient flow of the convex, lower semi-continuous functional iff the differential inclusion
is satisfied almost everywhere with respect to .
In practice for a given flow
if is differentiable and is given as an integral over some domain, say, , we simply verify there is some number so that
If , the flow is called positive gradient flow. Now we discuss some examples.
Example 1. The following semilinear heat equation
corresponds formally to the gradient flow associated to the energy functional
Here we assume the boundary and initial conditions are all zero just for simplicity.
is the gradient flow associated to the energy functional
Example 2. Heat flow for Nirenberg’s problem
is also a gradient flow for the following functional
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. -curvature flow
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.