# Questions tagged [gradient-flows]

An ordinary differential equation that generalizes the notion of "path of steepest descent." For questions on "gradients" of a function, use (multivariable-calculus) instead.

130 questions
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### Existence of a function with $||grad f||>\epsilon$

I want to construct a function $f$ on the unit ball $B$ of $\mathbb{R}^n$, such that it is negative on a closed subset of the boundary $\partial'B\subsetneqq\partial B$, zero on a given point $p\in B$,...
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### Legendre-Fenchel conjugate: $\dot x \in \operatorname{dom}(\Psi^*), -\operatorname{D}\mathcal E(x) \in \partial \Psi^*(\dot x)$

Consider $\mathcal E: \Bbb R \to \Bbb R$ lower semi-continuous with $\mathcal E(x) \to \infty$ if $|x| \to \infty$ and the equation $\dot x = -\nabla \mathcal E(x)$. Apparently, this differential ...
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### Level sets of function with constant sign partial derivatives

Consider a smooth function $f: E \to \mathbb{R}$ in the closed subset $E \subset \mathbb{R}^N$ with boundary $\partial E$. Show that if the partial derivatives $\partial f/\partial x_j$ do not ...
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### Volume Preserving Flow after Variable Change of Transport Equation

Suppose that we have the transport equation with non-consant coefficients in divergence from $$\partial_t f(t,x,\xi) + \nabla_{x,\xi}\cdot (F(x,\xi) f(t,x,\xi)) = 0 \\ f(0,x,\xi) = f_0(x,\xi)$$ where ...
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### Textbook Recommendations: Solving Systems of Matrix ODEs

This is in reference to the works of Trendafilov whose approaches to multivariate statistical problems boil down to solving a dynamical system involving matrices. Question: Can anyone suggest a book/...
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### Convergence of gradient flow

I'm interested in a simple property of the gradient flow $x'(t) = - \nabla f(x)$: under what conditions on $f$ does the gradient flow converge to a stationary point? In particular, I'm interested in ...
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### Asymptotic behaviour of gradient flows for $t \to \infty$

I have often heard about the asymptotics of gradient flows converging to some "equilibrium point" as $t \to \infty$. This concept has come to my ear by word of mouth multiple times and is often ...
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### 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 ...
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### Level sets of $C^1$ function on Riemannian manifold

Suppose a continuously differentiable function $f$ on a Riemannian manifold $M$, $f:M\rightarrow \mathbb{R}$, has a set of local minimum points $S$ with $f(x)=f(y)$, for all $x,y \in S$. Is it ...
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### Taking divergence of the gradient of a scalar field that depends only on the position vector in $\mathbb{R}^3$.

The scalar field $f$ depends only on $r=|\underline{\mathbf{r}}|$ which is the position vector in $\mathbf{R}^3$ and I need to calculate the quantity, $$\nabla \cdot \nabla f$$ i went about ...
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### why is my expectation of where the critical points are disagreeing with my graph?

given the parametrised function $$f_t(x,y,z)=z^3+tz-x^2+y^2$$ We know that that the gradient vector field is $\nabla f_t=(-2x,2y,3z^2+t)$ Now I originally thought that the critical points were at ...
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### Clarification on Theorem 4.1 Milnor's lecture on h-cobordism theorem. Why The set $K_p$ is compact?

I need a clarification about a passage in the statement of Theorem $4.1$ of Milnor's Lectures on the H-cobordism theorem Suppose that for some choice of gradient-like vector field $\xi$, the ...
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### a question about gradient

Let $f(x,y)=\ln\|\mathbf r\|$ where $\mathbf r=x\mathbf i+y\mathbf j$. Show that $\nabla f=\frac{\mathbf r}{\|\mathbf r\|^2}$. I attempt to calculate $\nabla\mathbf r$. But I have no idea how to ...
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### Conditions for gradient dynamical systems

In S. Smale's, “On gradient dynamical systems,” Ann. of Math. (2), vol. 74, no. 1, pp. 199–206, 1961, four conditions on a vector field $X$ on a compact manifold $M$ are given that are sufficient for ...
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### Help understanding how to compute Laplace operator?

I'm working on a diffusion simulation, and I am struggling with how to calculate the Laplace operator for my density function as it applies to Flick's second law. Conceptually I believe it is clear: ...
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### Cauchy-Riemann Reparametizing level curves to gradient flow lines Using Picard-Lindelof

Problems in question here How can I prove that level curves of function V can be reparametrized to gradient flow lines of U? Any help would be appreciated and if anybody has any tips for part 3 would ...
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### Drawing Vector Fields (beginner)

I have $f(x,y) = xy$ which has the Gradient Vector Field F$(x,y) = (y,x)$ I believe the correct way to draw individual vectors is to map points into the formula, for example at point $(1,-3)$ we will ...
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### How can entropy be concave in time?

At this point in a lecture by Cedric Villani, prof. Villani talks about his work revealing that "if entropy is a concave function of time, then the Ricci curvature is non-negative." But wait, how can ...
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### Normal of Implicit Surface

I am trying to understand the reasoning behind finding what the normal of an implicit surface is. I found this article on the web: http://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node27.html ...