  We study both the local and global existence of a gradient flow of the
Sinai-Ruelle-Bowen entropy functional on a Hilbert manifold of expanding maps
of a circle equipped with a Sobolev norm in the tangent space of the manifold.
We show that, under a slightly modified metric, starting from any initial
value, every trajectory of the gradient flow converges to the map with a
constant expanding rate where the entropy attains the maximal value. In a
simple case, we obtain an explicit formula for the flow's ordinary differential
equation representation. This gradient flow has a close connection to a
nonlinear partial differential equation: a gradient-dependent diffusion
equation.
