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### Dynamical formulation of the 2-Wasserstein distance for *discrete* matrix-valued measures

TL;DR: I want to find a definition generalizing "$t \mapsto \frac{1}{m} \sum_{k = 1}^{m} \delta_{x_k(t)}$ is a Wasserstein gradient flow" to matrix-valued probability measures. Let $(X, d)$ ...
146 views

### Gradient descent relaxation dynamics of a Euler-Lagrange equation

I want to minimize the functional $$F=\int{L(u)}dx,$$ where $L= u_x^2-u^2$ is the Lagrangian function of the functional. Even if its Euler-Lagrange equation is easily found and solved, I want to try ...
153 views

Consider a second order gradient-like system with linear damping $$\ddot{x}+\dot{x}+\nabla f(x)=0, \quad x(0)=x_0,\quad\dot{x}(0)=0$$ Suppose $f\in C^2(\mathbb{R}^n)$ and $\inf_{x\in\mathbb{R}^n}f(x)&... 3 votes 1 answer 110 views ### Equivalent definition of the Kantorovich-Fisher-Rao distance I am reading this paper "A JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows" (https://arxiv.org/abs/1602.04457) and in the proof of Proposition 2.2, basically, if the measure ... 3 votes 0 answers 70 views ### Regularity of center manifold Consider a$C^r$vector field$f \colon \mathbb{R}^n \to \mathbb{R}^n$with$r \geq 1$. Let$\bar x$be a critical point of$f$, that is,$f(\bar x) = 0$. Suppose that the spectrum of$\mathrm{D}f(\...
191 views

Is there a construction in Riemannian geometry which relates the gradient flow of a function on a manifold with a certain metric with geodesics on another related manifold with its own metric?
441 views

### The negative gradient flow of a Morse-Bott function on a compact manifold converges to a critical point?

Let $(M, g)$ be a compact Riemannian manifold and $f: M \rightarrow \mathbb{R}$ be a Morse-Bott function, i.e. the set a critical points of $f$, $Crit(f)$, has connected components which are smooth ...
164 views

Theorem: Suppose ${\bf g} :\mathbb{R}^n \mapsto \mathbb{R}^n$ is continuously differentiable, there exists a set $\mathcal{A} \subset \mathbb{R}^n$ such that $\bf g$ is uniformly Lipschitz on $\... 2 votes 0 answers 223 views ### Do Chern-Simons terms qualitatively alter the behavior of the Yang-Mills gradient flow? I'm reading about the Yang-Mills heat flow, and I'm curious how adding a Chern-Simons term alters its solutions. This is probably elementary or folklore, but I don't know well enough to say. ... 0 votes 1 answer 314 views ### Rewriting PDE as "push-forward" Suppose that we have the following PDE $$\partial_t \mu_t = \nabla\cdot \left(\nabla \mu_t - (b*\mu_t)\mu_t\right), \tag{1}$$ with$\mu_0$being a (smooth) probability measure/density on$\mathbb{R}^d$... 3 votes 1 answer 369 views ### Geometric flow by the level sets of a harmonic function Let$u$be an harmonic function in a cylindrical domain$B_2^{n-1}\times(-1,1)\subset\mathbb{R}^n$, and suppose its level sets$\Gamma_t=\{u=t\}$are graphs of functions on$B_2^{n-1}$. Consider a ... 2 votes 0 answers 56 views ### Chain recurrent points of a gradient-like system Let$X$be a compact metric space and$f:X\to X $homeomorphism. Let$V:X\to \mathbb{R}$be a Lyapunov function for$(X,f)$(continuous function such that$(\forall x\notin Fix(f))\ \ V(f(x))<V(x))...
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I've tried asking this question on Mathematics site, but I only got an upvote and no answer. I've searched online, tried to find something about this topic, but I haven't found much (and the things I ...
189 views

### Intersection of self-shrinkers

I have a problem regarding a statement in the paper Smooth compactness of self-shrinkers by Colding and Minicozzi. In the article, they define a surface $\Sigma$ in $\mathbb R^3$ to be a self-shrinker ...
118 views

### What kinds of gradient-flows on $\mathbb R^d$ preserve the log-concavity of the distribution $\mu_0$ of starting point $x_0$

Let $\mu_0$ be a log-concave distribution on $\mathbb R^d$ and let $f:\mathbb R^d \to \mathbb R$ be $C^2$. Let $x_0$ be sampled uniformly at random from a log-concave distribution $\mu_0$, meaning ...
47 views

### A question about strong slopes (nonsmooth analysis)

Context. I'm reading the manuscrip "Nonlinear Error Bounds via a Change of Function" by Dominique Azé and Jean-Noël Corvellec (J Optim Theory Appl 2016), and I'm having a hard time ...
142 views

### Metric obstructions for area-preserving diffeomorphisms with constant singular values

Let $\mathbb{T}^2$ be the topological $2$-dimensional torus, and let $0<\sigma_1 < \sigma_2$ satisfy $\sigma_1 \sigma_2=1$. Let $g$ be an arbitrary smooth Riemannian metric on $\mathbb{T}^2$. ...
1 vote