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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} -convex, which generalizes the notion of convexity. We have omitted that discussion for the sake of clarity and brevity. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is indeed convex, then the subdifferential becomes,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t):(0,\infty)\rightarrow\text{dom}(f)} is a locally absolutely continuous function with the property that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x'(t)\in\partial(f(x(t))} for almost every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} (with respect to Lebesgue measure). Note that the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t)} being locally absolutely continuous is necessary for the existence (almost everywhere) of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x'(t)} . It will be particularly useful to identify the starting point of a gradient flow Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t)} , which is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0:=\lim_{t\rightarrow 0}x(t)} .

Main Existence Theorem

From Linear to Nonlinear Operators

Recall that the Hille-Yosida theorem states the following:

Theorem Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} be a densely defined linear operator on a Banach space (note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} need not be bounded, but we assume Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is closed). Denote Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho(T):=\left\lbrace z\in\mathbb{C}:(z-T) \text{ is invertible and bounded}\right\rbrace} as the resolvent set of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} . Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is the infinitesimal generator of a semigroup Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_t} if and only if, for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda>0} , we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda\in\rho(T)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ||(\lambda-T)^{-1}||\leq\frac{1}{\lambda}}

Using this result, we may view a linear time evolution PDE as a Banach space valued problem of the following form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi'(t)=T\phi }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(0)=\phi_0 }

which has solution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(t):=S_t\phi_0} . In the ODE above, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} denotes the linear differential operator in the original linear time evolution PDE.

This approach works well when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is linear, but requires some significant modification in the case that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (I+\epsilon T)^{-1}} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is a non-linear operator on a Hilbert Space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} . Moreover, the existence argument in classical proof the Brézis-Komura Theorem requires a modification to the generalized resolvent of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} . This modification is the Yosida Regularization,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y(T,\epsilon):=\frac{I-(I+\epsilon T)^{-1}}{\epsilon}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} is some non-negative parameter. The Yosida Regularization, being Lipschitz as an operator on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} with Lipschitz constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2}{\epsilon}} , allows one to construct the starting point for a solution in the Brézis-Komura Theorem. This is analogous to the Moreau-Yosida Regularization, where lower semicontinuous function is approximated by a Lipschitz function.

Another way to see the Yosida Regularization is via a time-discretization, the approach first used by Jordan, Kinderlehrer and Otto. There are two ways of discretizing a gradient flow, namely Euler schemes, explicit one:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{X_{t}^{n+1}-X_{t}^{n}}{t} = - \nabla f(X_{t}^{n}) } ,

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_{t}^{0} = X^{0} } , and implicit one:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{X_{t}^{n+1}-X_{t}^{n}}{t} = - \nabla f(X_{t}^{n+1}) } ,

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_{t}^{0} = X^{0}. }

Explicit one is easier for implementation, but the implicit one is more natural here since it decreases, same as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x(t)). }

In order to formulate the Brézis-Komura Theorem, we introduce $\lambda$- convexity:

Definition Given $\lambda \in \mathbb {R} ,$ we say that $f:H\rightarrow (-\infty ,\infty ]$ is $\lambda$- convex if $f-{\frac {\lambda }{2}}|\cdot |^{2}$ is convex.

The Brézis-Komura Theorem

We restate the Brézis-Komura Theorem as is stated in Ambrosio et al.

Theorem Assume that $f$ is $\lambda$-convex for some $\lambda \in \mathbb {R}$ and lower semicontinuous. For every $x_{0}\in {\overline {{\text{dom}}(f)}}$, there exists a unique gradient flow $x(t):=S_{t}x_{0}$ starting at $x_{0}.$ The family of operators $\left\lbrace S_{t}\right\rbrace _{t>0},$ satisfies the semigroup property $S_{t+s}=S_{t}\circ S_{s},$ and the contractivity property

$|S_{t}x_{0}-S_{t}y_{0}|\leq e^{-\lambda t}|x_{0}-y_{0}|\quad \forall x_{0},y_{0}\in {\overline {{\text{dom}}(f)}}.$

Example and Applications

As suggested by our previous discussion, the Brézis-Komura Theorem may be used to assert the existence of flows solving certain nonlinear time-evolution PDEs. Several nonlinear time-evolution PDEs and their solutions are discussed by both Ambrosio et al. and Evans. A simple example consists of a particular case of the so-called $p$-Laplace equation on $L^{2}(\mathbb {R} ^{n})$, which seeks to find a solution to the heat-like equation $u_{t}-\nabla \cdot (|\nabla u|^{2}\nabla u)=0$. Motivated by the applicable variational formulation of this problem, one may consider the function $T(u):=\int _{\mathbb {R} ^{n}}{\frac {|\nabla u|^{4}}{4}}$ whenever $u\in L^{2}(\mathbb {R} ^{n})\cap W^{1,4}(\mathbb {R} ^{n})$, with $T(u):=\infty$ otherwise. Applying the Brézis-Komura Theorem yields a flow $x(t)$ such that $x'(t)=\nabla \cdot (|\nabla x|^{2}\nabla x)$. Some care must be taken to show that the subdifferential of $T$ coincides with the right hand side of the expression for $x'(t)$.

In general, the $p$-Laplace equation is given by $u_{t}-\nabla \cdot (|\nabla u|^{p-2}\nabla u)=0$ and is solved on $L^{2}(\mathbb {R} ^{n})$. Note that the $p$-Laplace equation is a generalization of the heat equation and we may recover the heat equation on $L^{2}(\mathbb {R} ^{n})$ when $p=2$. In that case, the Brézis-Komura Theorem may be applied to the function $T(u):=\int _{\mathbb {R} ^{n}}{\frac {|\nabla u|^{2}}{2}}$ whenever $u\in H^{1}(\mathbb {R} ^{n})$, with $T(u):=\infty$ otherwise. Thus, we acquire the existence of a gradient flow which satisfies the heat equation.

The Brézis-Komura Theorem is also used to assert the existence of the Riemannian Heat Semigroup. This forms the starting point for connecting optimal transport and ricci curvature.