An invariance principle for gradient flows in the space of probability measures
This article studied the qualitative convergence results of a general class of evolution PDEs described by p-Wasserstein gradient flows. These qualitative convergence results come from dynamical systems known as the LaSalle Invariance Principle. A weaker form of this principle is a well known result in finite dimensional gradient flows: Let $F: \mathbb{R}^d \to \mathbb{R}$ be a sufficiently smooth function and $x: [0 , \infty)\to \mathbb{R}^d$ be a curve solving the ODE \[ \frac{d}{dt}x(t) = -\nabla F(x(t)), \ \ x(0) = x_0 \in \mathbb{R}^d. \] Assume further that the set $\{x(t) : t \geq 0\}$ is relatively compact in $\mathbb{R}^d$. Then, the distance between $x(t)$ and the set of critical points of $F$ (i.e. $\{ y \in \mathbb{R}^n \mid \nabla F(y) = 0\}$) converges to $0$ as $t \rightarrow \infty.$ The authors studied this principle in the context of gradient flows in the metric spaces $(P(\Omega),W_p)$ and $(P_p(\Omega), W_p)$ where $\Omega$ is either $\mathbb{R}^d$ or the torus $ \mathbb{T}^d$ and $P(\Omega)$ (respectively, $P_p(\Omega)$) is the set of all Borel probability measures on $\Omega$ (respectively, with finite $p$-th moment) and $W_p$ denotes the $p$-Wasserstein distance defined by \[ W_p(\mu, \nu)^p = \inf \left\{ \int_{\mathbb{R}^d\times \mathbb{R}^d} |x-y|^p \ d\pi(x,y) \mid \pi \in P(\mathbb{R}^d \times \mathbb{R}^d)\text{ has marginals }\mu\text{ and }\nu\right\}. \] There are multiple notions of gradient flows on a general metric space: \begin{itemize} \item[(a)] $p$-curves of maximal slope. This is a very weak notion founded on an appropriately defined metric derivative of a curve $\rho(t): [0,\infty)\to P_p(\Omega),$ weak upper gradient of the energy $E : P_p(\Omega) \to \mathbb{R} \cup \{+\infty\}$ and Young's inequality. In this formalism, it is rather easier to establish the existence theory but does not guarantee the uniqueness of solutions. \item[(b)] Evolutionary Variational Inequality (EVI), which requires a notion of convexity assumption on the energy $E$, existence theory is relatively difficult to establish but guarantees the uniqueness of solutions. \end{itemize} When $p=2,$ \textit{F. Otto}'s [Commun. Partial Differ. Equations 26, No. 1--2, 101--174 (2001; Zbl 0984.35089)] formal Riemannian calculus on $(P_2,W_2)$ provides us with a nice formula for $2$-curve of maximal slope and it is given by \[ \partial_t \rho = \nabla \cdot (\rho \nabla E^{\prime}(\rho)) \] where $E: P_2(\Omega) \to \mathbb{R}\cup \{+\infty\}$ can be thought of the energy driving the system. Based on the seminal work of \textit{L. Ambrosio} et al. [Gradient flows in metric spaces and in the space of probability measures. Basel: Birkhäuser (2005; Zbl 1090.35002)] on curves of maximal slope in $(P_p,W_p)$ and by combining some of the basic notions of dynamical systems, the authors reproduce LaSalle invariance principle in this context. They apply this abstract theory to several PDEs which include (a) Fokker-Planck equation, (b) The noisy Kuramoto model on the unit circle, (c) McKean-Vlasov equation, (d) Porous medium equation, (e) Degenerate diffusion equations on the torus, (f) Aggregation-diffusion equation of porous medium type and (g) Nonlocal interaction equations describing a model for consensus convergence. As a consequence, the authors recover, simplify, and even extend the results in their respective literatures. The article ends with two appendices (i) a brief introduction and preliminaries from the theory of the continuous-time metric dynamical systems and (ii) a general lower semicontinuity result for a class of weak upper gradients in $(P_2,W_2).$
scientific article; zbMATH DE number 7633392
scientific article