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 inclusionis 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.

Observe that

and

.

Therefore

i.e.

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.

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