Gradient flows are an important subclass of evolution equations. A gradient-flow structure can be exploited to obtain additional information about the evolution, such as existence and the stability of solutions. Moreover, gradient flows can provide additional physical and analytical insight, such as the maximum dissipation of entropy and energy, or the geometric structure induced by the dissipation distance.
A special subclass is formed by gradient flows with respect to the Wasserstein distance. This class was first identified in the seminal work by Jordan, Kinderlehrer, and Otto in the late nineties. In my talk I will introduce the Wasserstein distance and discuss its relation to diffusion equations.