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 results in the Hille-Yosida theorem, which gives necessary and sufficient conditions for an 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, rewritten from Ambrosio et al.'s definition.

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 et al. discusses the case when $f$ is $\lambda$-convex, which generalizes the notion of convexity. We have omitted that discussion for the sake of clarity and brevity. 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 $x'(t)\in \partial (f(x(t))$ for almost every $t$ (with respect to Lebesgue measure). Note that the $x(t)$ being locally absolutely continuous is necessary for the existence (almost everywhere) of $x'(t)$. It will be particularly useful to identify the starting point of a gradient flow $x(t)$, which is given by $x_{0}:=\lim _{t\rightarrow 0}x(t)$.

Main Existence Theorem

From Linear to Nonlinear Operators

Let $T$ be a densely defined linear operator on a Banach space (note that $T$ need not be bounded). Recall that the Hille-Yosida theorem asserts that $T$ is the infinitesimal generator of a semigroup $S_{t}$ if and only if, for each $\lambda >0$, we have $\lambda \in \rho (T)$ and $||(\lambda -T)^{-1}||\leq {\frac {1}{\lambda }}$ . Here, $\rho (T)$ denotes the resolvent set of $T$. Using this result, we may view a linear time evolution PDE as a Banach space valued problem of the following form:

$\phi '(t)=T\phi$

$\phi (0)=\phi _{0}$

which has solution $\phi (t):=S_{t}\phi _{0}$ . In the ODE above, $T$ denotes the linear differential operator in the original linear time evolution PDE.

This approach works well when $T$ is linear, but requires some significant modification in the case that $T$ is nonlinear. In particular, observe that the Hille-Yosida theorem makes use of the resolvent of the relevant linear operator. In some sense, the resolvent bypasses the problems that arise from the unboundedness of the linear operator; in particular, it makes sense to discuss power series expansions involving the resolvent. In the unbounded case, one must introduce a generalization of the resolvent.

In the notation of Ambrosio et al., the analogue of the resolvent used in the proof of the Brézis-Komura Theorem is $(I+\epsilon T)^{-1}$ where $T$ is non-linear. Moreover, the existence argument in the Brézis-Komura Theorem requires a modification to the generalized resolvent of $T$. This modification is the Yosida Regularization,

${\frac {I-(I+\epsilon T)^{-1}}{\epsilon }}$

where $\epsilon$ is some non-negative parameter. The Yosida Regularization, being Lipschitz with Lipschitz constant depending on $\epsilon$, allows one to construct the starting point for a solution in the Brézis-Komura Theorem.