Gradient flows in Hilbert spaces

Gradient Flows in Hilbert Spaces are generalizations of time-derivatives with a gradient constraint. Specifically, a gradient flow is a Hilbert Space valued function who's time derivative lies in some generalized collection of gradient vectors. Gradient flows are a key topic in the study of non-linear time evolution partial differential equations. In this exposition, we will draw from Ambrosio et al.'s resource Lectures on Optimal Transport and Evans' Partial Differential Equations

Introduction

The heat equation is a classic example of a time evolution partial differential equation. In particular, the heat equation is a linear parabolic partial differential equation. Such PDEs are well understood and are solvable using several different approaches. One particularly interesting technique is to view the PDE as a Banach-space valued ODE in the time variable. In this case, we can try to understand how to write the solution of the PDE as a flow in time which is a generalization of the exponential function. The techniques which one implements to find such a solution ultimately result in the Hille-Yosida theorem, which gives necessary and sufficient conditions for the operator T to be infinitesimal generator of a contraction semigroup of the given PDE. In some sense, these ideas can be extended to non-linear time evolution PDEs, leading to the general notion of flows on Hilbert spaces. We will discuss in this article how the theory of flows can be used to yield existence of a solution of a "non-linear heat equation."

Definitions

Let $H$ be a Hilbert space with inner product $\langle \cdot ,\cdot \rangle$ with induced metric $||\cdot ||$. Throughout this exposition, we assume that $f:H\rightarrow (-\infty ,\infty ]$ is proper, so that the domain on which it takes finite values, ${\text{dom}}(f)$, is not empty.

First, we recall the notion of the subdifferential. Ambrosio discusses these ideas in more generality than are discussed here; we restrict to more elementary cases for the sake of brevity and clarity.

The subdifferential of $f$ at $x\in {\text{dom}}(f)$ is the collection,

$\partial (f(x)):=\left\lbrace v\in H:f(u)\geq f(x)+\langle v,u-x\rangle +o(||u-v||)\right\rbrace$

Remark: observe that we are not assuming $f$ is convex, only that it is proper. In fact, Ambrosio discusses the case when $f$ is $\lambda$-convex, which generalized the notion of convexity. If $f$ is indeed convex, then the subdifferential becomes,

$\partial (f(x)):=\left\lbrace v\in H:f(u)\geq f(x)+\langle v,u-x\rangle \quad {\text{for each }}u\in H\right\rbrace$

A gradient flow $x(t):(0,\infty )\rightarrow {\text{dom}}(f)$ is a locally absolutely continuous function with the property that $x'(t)\in \partial (f(x(t))$.