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

.

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