# Questions tagged [flows]

The flows tag has no usage guidance.

92
questions

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### Cocycle-conjugacy classes of flows on the C*-algebra of compact operators

A flow on a C*-algebra $A$ is a group homomorphism $\sigma $ from ${\mathbb R}$ into the group of *-automorphisms of $A$
such that the map
$$
t\in {\mathbb R}\mapsto \sigma _t(a)\in A
$$
is norm-...

1
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1
answer

96
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### Singularities of mean-convex MCF in the sphere?

Let $\Sigma^n \subset S^{n+1}$ be a codimension one, embedded minimal hypersurface in the sphere. As the sphere has positive Ricci curvature, this must be unstable. In particular, perturbing $\Sigma$ ...

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votes

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answers

67
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### Trapped vs. nonwandering points

For a continuous flow on a topological space the concept of a nonwandering point (in forward/backward time) is well-known. Another useful concept (for example from the point of view of scattering ...

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### Methods for calculating (one-parameter subgroup) actions

For $G$ a Lie group and $\mathfrak{g}$ its Lie algebra, I am interested in one-parameter subgroup actions on “functions” $f$ of the form
\begin{equation}
\mathrm{e}^{t L(z)} f(z)
\end{equation}
...

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0
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### Singular asymptotic limits of mean-convex MCF

Let $(M_t \mid t \geq 0)$ be a mean-convex mean curvature flow of hypersurfaces in ambient Riemannian manifold $(N^{n+1},g)$. Brian White proved that this flow (defined 'weakly' as a level set flow ...

4
votes

1
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151
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### One-sided version of the curve-shortening flow

The curve-shortening flow is
$$
\frac{\partial C}{\partial t} = \kappa n
$$
where $\kappa$ is the curvature, and $n$ is the unit normal vector. For a smooth Jordan curve $C\subset\mathbb R^2$ (closed ...

0
votes

1
answer

122
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### Distance function to mean curvature flow

In Ilmanen's monograph on elliptic regularization and mean curvature flow, the following claim is made. (Just for reference, this is on page 60.) For the proof—apparently standard calculations—the ...

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### Vector fields $X$ and $Y$ commute on a closed set $K$. Do there exist commuting $\tilde X,\tilde Y$ with $\tilde X=X$ and $\tilde Y=Y$ on $K$?

I have a nice research idea whose proof hinges on the following question
Suppose $X_p$ and $Y_p$ are vector fields in $\mathbb{R}^3$ with $[X,Y]_p=0$ for all $p$ in some closed set $K\subset\mathbb{R}...

5
votes

0
answers

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### Metric under Ricci flow on a 2-sphere can be realized by embedding

I am sorry if this is a silly question, but I am new to Ricci flows.
Let $\Sigma \subset \mathbb{R}^3$ be a smoothly embedded sphere, and denote its metric (induced by $\mathbb{R}^3$) by $g$. Suppose ...

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votes

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124
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### Choosing the derivative of a flow

I am looking for something like the Franks' Lemma for flows. The celebrated Franks' Lemma states that: Let $f:M \rightarrow M$ be a $C^1$ diffeomorphism and $S=\{p_1,...,p_k\}$ be a finite set of ...

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votes

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379
views

### Kneser theorem about the Klein bottle

I know that in $1923$ H. Kneser showed that a continuous flow in a Klein bottle without singular points has a periodic trajectory. The original article is this, but does anyone know another old or new ...

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0
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48
views

### Existence of measure-preserving Lagrange flow for inhomogeneous transport equation

I asked this question on stackexchange:
Let us consider the Cauchy problem for the transport equation
$$ \partial_t \varphi + b\cdot \nabla \varphi= f \text{ in } (0,T)\times\mathbb{R}^3,\\ \varphi(0,...

1
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66
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### Hyperplanes which equalize the Radon transforms of two distributions

Let $p_1$ and $p_2$ be "nice" probability densities on $\mathbb R^m$, for example the densities of a multivariate Gaussians $N(\mu_1,\Sigma)$ and $N(\mu_2,\Sigma)$ with common covariance ...

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67
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### $\mathbb{R}^n$-flow, cross-section and Whitney theorem

For a $\mathbb{R}$-flow (X, $\Phi_{\mathbb{R}}$), the (local) cross-section is well defined (recall that a subset $S\subset X$ is a cross section of time $\xi>0$ if $S\cap \Phi_{[-\xi, \xi]}(x)=\{x\...

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68
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### Is there a theory for forcing "sludge" through a network, analogous to electric current flows?

I'm familiar with the correspondence between reversible Markov chains, random walks, and electric current flows, as described in Probability on Trees and Networks by Lyons and Peres.
Is there an ...

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votes

2
answers

304
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### Planar flow with bounded orbits and a single equilibrium point

Is there a $C^1$ flow $\varphi_t$ defined on ${\bf R}^2$ with a single fixed point $0$ and such that for all x,
$$\lim_{t\rightarrow +\infty}\varphi_t(x) = 0,$$
$$\lim_{t\rightarrow -\infty}\varphi_t(...

1
vote

1
answer

109
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### Is the union of two proper flows proper?

Let $\varphi _t$ be a flow, aka. a one parameter group of homeomorphisms of the open subset $\Omega \subseteq {\mathbb R}^n$, which we assume to be continuous in the usual sense
that
$$
(t, x)\in {\...

7
votes

1
answer

168
views

### Stability Question for Isotopies Between Compact Sets

Suppose $X, Y$ are compact sets in $\mathbb{R}^2$ and $F$ is an ambient isotopy carrying $X$ onto $Y$.
Is there an ambient isotopy $F'$ agreeing with $F$ on $X$ and which is constant in a ...

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answers

93
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### Existence theory for geometric flow of space curves

Is there any existence theory applicable to general geometric flows of space curves in the following form?
$$
\partial_t \gamma = v_t t + v_n n + v_b b
$$
Here $\gamma$ is the evolving curve, $t$, $n$ ...

2
votes

0
answers

207
views

### Show that the manifold interior is invariant under this flow

Let $\tau>0$, $d\in\mathbb N$, $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be continuous in the first argument with $$\sup_{t\in[0,\:\tau]}\left\|v(t,x)-v(t,y)\right\|\le c\left\|x-y\right\|\tag1\;\...

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### Entropy of flow and time-1 map

Let $\Phi=(\phi_t)_{t\in \mathbb{R}}$ be a continuous flow on a compact metric space $X$. Let $\mu$ be a $\phi_1$-invariant measure. Then it is not hard to verify tht $\int_{0}^{1} \phi_t\mu dt$ is $\...

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125
views

### Divergence as infinitesimal volume change on a Finsler manifold

Let $M$ be a smooth manifold and $Z$ a smooth vector field on it.
It generates a family of diffeomorphisms $\phi_t:M\to M$ by demanding that $\phi_0=\operatorname{id}$ and $\partial_t\phi_t(x)=Z(\...

3
votes

1
answer

269
views

### Conditions on the velocity ensuring that a flow moves points along the boundary of a manifold

Let
$\tau>0$;
$d\in\mathbb N$;
$v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be Lipschitz continuous in the second argument uniformly with respect to the first with $v(\;\cdot\;,x)\in C^0([0,\tau],\...

1
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0
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74
views

### Show that the support of the shape gradient $\nabla\mathcal F(\Omega)$ is contained in $\overline\Omega$

Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $(T^{(\theta)}_t)_{t\ge0}$ denote the $C^1$-diffeomorphism from $E$ onto $E$ with $$T^{(\theta)...

2
votes

1
answer

414
views

### Flow induced by differentiable velocity field is differentiable

Let $E$ be a $\mathbb R$-Banach space, $\tau>0$ and $v:[0,\tau]\times E\to E$ such that$^1$ $$x\mapsto t\mapsto v(t,x)\tag1$$ belongs to $C^{0,\:1}(E,C^0([0,\tau],E))$. This is enough to ensure ...

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votes

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### What is the importance of singularities of type II in the Mean Curvature Flow?

I am reading the Mean Curvature Flow and Isoperimetric Inequalities by Manuel Ritoré, Carlo Sinestrari, Vicente Miquel and Joan Porti and I am curious to know what is the importance in understand the ...

1
vote

1
answer

93
views

### Explicit transitive flow on disc

$D_n\triangleq \left\{x \in \mathbb{R}^n:\, \|x\|\leq 1\right\}$ with its subspace topology. By a transitive flow on $D_n$ I mean a continuous function
$$
\phi: [0,1]\times D_n\rightarrow D_n,
$$
...

2
votes

0
answers

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### If a stochastic flow is Fréchet differentiable in the spatial parameter, does the induced transition semigroup preserve differentiability?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space, $X:\Omega\times[0,\infty)\times E\to E$ be $(\mathcal A\otimes\mathcal B([0,\infty))\otimes\...

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votes

1
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260
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### Smoothening pseudo-Anosov flows

A topological Anosov flow on a closed 3-manifold can be replaced by a smooth Anosov flow using an argument of Fried: use Markov partitions to find a surface of section, put in other terms, one can ...

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0
answers

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### Results on: (path/initial condition)-dependent variant of exponential map generates compactly supported diffeomorphisms

Let $M$ be a connected and simply connected Riemannian manifold, non-compact, and suppose that $\{V_p\}_{p \in M}$ is a family of vector fields on $M$ indexed by $M$. Suppose moreover that the map
$$
...

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0
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88
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### Convergence rate of Toda/Morse flow

Let $A(t), A_0$ be a $n\times n$ hermitian complex matrices and consider the following matrix flow
\begin{align}
\frac{dA}{dt} &= \left [ C\circ A , A \right ] \\
A(0) &= A_0 \ .
\end{align}
...

2
votes

0
answers

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### Definition of Lie derivatives of sections of natural vector bundles - product preservation needed?

Section 6.15 of Natural Operations by Kolár, Michor, and Slovak defines the Lie derivative of a section of a natural vector bundle along a vector field. Set-theoretically, the definition is clear. ...

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votes

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### Poking into a Lie group with your finger

I consider this as a differential geometry problem. I have asked some
of my classmates who are more interested in that, and also looked into
some literature, but none of what I've found seems to help.
...

0
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0
answers

463
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### directional derivative along geodesic flow of vector field

A rather elementary question for the differential geometers. Let $M$ be a Riemannian manifold, let $X\colon M \to TM$ be a vector field, and let $$\phi_t = \exp(tX)$$ be the "geodesic flow" of the ...

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votes

2
answers

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views

### Formal mathematical definition of renormalization group flow

I was watching some lectures by Huisken where he mentioned that one-loop renormalization group flow was in some analogous to mean curvature flow. I have tried reading up the exact definition of what ...

3
votes

2
answers

350
views

### Proof of Isoperimetric Inequality using Curve Shortening Flow

I am aware that there is a nice result where one uses curve shortening flow to prove the isoperimetric inequality on the Euclidean plane, but could not seem to find the original article where this was ...

2
votes

0
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94
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### Solution of equation on vector field

I have a vector field function $\vec{J}: {\bf R}^3\to {\bf R}^3$ looking like:
$$ \vec{J}(\vec{r}) = (\vec{B} \times \vec{v}(\vec{r}))\rho(\vec{r}) $$
with a (very well behaved) real, positive, ...

5
votes

1
answer

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### Modulus of continuity of flow for non-Lipschitz vector fields satisfies Osgood condition

An Osgood modulus of continuity is an increasing function $\omega:(0,1]\to(0,1]$ such that $\int_0^1\frac{dt}{\omega(t)}=\infty$.
We say a vector field $X$ satisfies Osgood condition with modulus $\...

4
votes

0
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### Geometric meaning of the extrinsic curvature neck

I'm reading by myself "Mean curvature flow with surgeries of two–convex hypersurfaces" (it's free available, it's just click on "Download PDF" on the upper right corner side) by Gerhard Huisken and ...

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### Characterization of Time-homogeneous flows for conditional expectation

Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...

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0
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### Ricci flow preserves locally symmetry along the flow

Let $(M,g_0)$ be a closed locally symmetric Riemannian manifold and let $g(t)_{t\in[0,T)}$ be a solution to the Ricci flow on $M$ with $g(0)=g_0$. How one can prove that Ricci flow preserves locally ...

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votes

1
answer

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### Convergence of Stochastic Flow but not Flow

Suppose that $\phi(t,x):[0,\infty)\times \mathbb{R}^d\rightarrow \mathbb{R}^d$ is a flow. Is it possible to extend $\phi$ to a family of stochastic flows $\{\Phi(t,x,\sigma)\}_{\sigma \in [0,1]}$ ...

4
votes

1
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325
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### Understanding the Hamilton's definition of $\ast$-operation

I'm studying by myself Mean Curvature Flow and I'm trying understand the definition of $\ast$-operation given by Richard Hamilton in the beginning of the section $13$ (page $40$) of his paper "Three-...

7
votes

1
answer

377
views

### Gradient flows on Hilbert manifolds

I would like to know if gradient flows of Morse-Bott functions on a Riemannian manifold always converge towards a unique critical point, provided that the flow line is bounded.
To be more precise, a ...

4
votes

0
answers

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### Approximation argument in geometric flows

I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken and I'm stuck in the following ...

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### Matching Stochastic Flows

Let $\nu = (\nu_t)_{t \in [0,T]} \in C( [0,T], \mathcal{P}_2(\mathbb{R}) ) ,$ where $\mathcal{P}_2(\mathbb{R})$ denotes the space of probability measures with finite moment equipped with the ...

3
votes

0
answers

202
views

### Interior gradient estimate of mean curvature equation

I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken. I'm stuck in the ...

0
votes

2
answers

94
views

### Does Max Flow produce uniform results? [closed]

I am interested in using Max Flow algorithm. I want to simulate transfer of quantity. Anyway, I am unsure of some thing.
Does Max Flow algorithm produce uniformly distributed max flow?
I have ...

2
votes

0
answers

240
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### Research trends on mean curvature flow

It's been a few months since I've been curious about mean curvature flow and now I'm reading Robert Haslhofer's lecture notes. I like the subject and would like to do research on it, but I know ...

7
votes

1
answer

779
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### Weak parabolic maximum principle on Riemannian manifolds

I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken, specifically, I'm ...