11
votes
"Wild" solutions of the heat equation: how to graph them?
I know this is a late answer, but I think that the other answers posted so far completely fail to illustrate the wildness of the function in question, since they plot it on a domain where not much ...
- 511
10
votes
Accepted
Gaussian distribution, maximum entropy and the heat equation
Both the Gaussian maximum entropy distribution and the Gaussian solution of the diffusion equation (heat equation) follow from the central limit theorem, that the limiting distribution of the sum of i....
- 188k
10
votes
Accepted
Linear PDE, analytic continuation, Green's function and boundary conditions
Q: Do I have to consider both problems (real $\xi$ or imaginary $\xi$) totally independently and work hard twice?.
A: A single calculation suffices, you could just do the inverse Fourier transform of $...
- 188k
9
votes
Accepted
$L^p$-norm under the heat flow
If I didn't do any miscalculations I believe I have proven the case $1\leq p\leq 2$. I will write $u$ instead of $u_t$. Let $(p)_k=p(p-1)\ldots(p-k+1)$ and
$$
w=\begin{pmatrix}
pu^{p-1}\Delta^2 u\\ (p)...
- 883
9
votes
Accepted
Unique solutions to the heat equation on $\mathbb{R}^3$
Sorry, maybe my previous comment was not clear enough. You have $$\frac{d}{dt} \int |u|^2 e^{-|x|}\, dx=-2\int |\nabla u|^2 e^{-|x|}\, dx +2 \int u \nabla u \cdot \frac{x}{|x|}e^{-|x|}\, dx.$$
Now use ...
- 6,035
9
votes
Accepted
Ricci flow negative curvature
The answer to your first question is Yes. The equation
$$
\tag{*} \partial_t g = -2\textrm{Ric} - 2(n-1)g
$$
is a Ricci flow type equation that admits hyperbolic space as a static solution. In fact, ...
- 7,197
8
votes
Accepted
The solutions of the heat equation from $0$ datum
Have you tried the classical Tychonoff's example, $$u(t,x) = \sum_{k = 0}^\infty \frac{g^{(k)}(t) x^{2k}}{(2k)!},$$ with $g(t) = e^{-1/t^\alpha}$ and $\alpha > 1$?
As discussed, for example, in ...
- 17.2k
7
votes
Gevrey estimate of derivatives
Faa di Bruno's formula for derivatives of compositions of functions says
$$
(f\circ g)^{(n)}(t)=n!\sum_{k\ge 0}\ \sum_{n_1,\ldots,n_k\ge 1}
\mathbf{1}\left\{\sum_{i=1}^{k} n_i=n\right\}
\ \frac{f^{(k)}...
7
votes
One question about the $\eta$ invariant
Summary: The integral $\int_0^\infty t^{\frac{s-1}{2}} \operatorname{Tr}[ D^u (\exp(-t D^2) ] \, dt$ has a simple pole at $s=0$ with residue $2 C_{-1/2}$, from which your expression for $\eta(0)$ ...
- 266
7
votes
Explicit solution of a free boundary problem for heat equation
From the maximum principle, the solution $v$ of
$$v_t=v_{xx}+1,$$
together with the data $v=0$ at the boundary and initial time, is positive. Therefore your $u$ is nothing but $v$. It turns out that ...
- 52.3k
6
votes
Accepted
Long-time decay of heat kernel on compact manifolds
The paper of Carlen , Kusuoka and Stroock relates Nash inequality to diagonal bounds for the Heat Kernel for both short times and all time.They also explain a method of E B Davies to convert diagonal ...
- 4,131
6
votes
Accepted
Possible flaw in the proof of the Eells-Sampson theorem on harmonic maps in Nishikawa's book
$\newcommand{\R}{\mathbb{R}}\newcommand{\pa}{\partial}$Edit: The answer is now LaTeXified.
Below are my notes on this. I reworked the proof:
Proof. Let $S:=\big\{T\in[0,\infty):$ the equation has a ...
- 365
6
votes
Accepted
McKean-Singer formula in Heat Kernels and Dirac Operators book
The assertion is supposed to be that $d(e^{-tD^2})/dt$ has the same smooth kernel as $-D^2 e^{-tD^2}$, i.e. they are the same operator. This is because $e^{-tD^2}$ is the solution operator to the ...
- 29.2k
6
votes
Accepted
Compactness for initial-to-final map for heat equation
Parabolic regularity show that $u$ is regular for all positive times; in particular $u(t,\cdot) \in W^{1,2}(M)$ for all $t > 0$. Interior parabolic estimates additionally show that there is a ...
- 5,038
6
votes
Accepted
Let $g$ be the heat kernel. Are there constants $C_1, C_2>0$ such that $\frac{g(t_1, \cdot)}{t_1} \le C_1 \frac{g(C_2 t_2, \cdot)}{\sqrt{t_2}}$?
The answer is no. E.g., let $t_1\sim t_2\downarrow 0$ and $|x|\sim\sqrt{t_2}$.
- 128k
6
votes
Accepted
How to get $\int_{\mathbb R^d} |\partial_i\partial_j(1-\Delta)^{-\frac{\delta}{2}}p_t(\cdot-y)(x)| \, \mathrm d x \lesssim t^{\frac{\delta}{2}-1}$?
This is a duality argument (the author is really invoking the adjoint of Lemma 5.2(2), rather than Lemma 5.2(2) directly). We can write
$$ I = \sup_g \left|\int_{{\bf R}^d} \partial_i \partial_j (1-\...
- 114k
5
votes
Accepted
Maximum principle for heat equation, low regularity case
This version is still true: if $u$ had a local maximum at $(x,\,T)$, say with $u(x,\,T) = 0$, then $u \leq 0$ in a small parabolic cylinder centered at $(x,\,T)$. After rescaling we can assume that $u ...
- 4,899
5
votes
Gaussian distribution, maximum entropy and the heat equation
Besides the central limit theorem, there is the connection between diffusion and Wasserstein distance $W_2(p,q)$ (the minimum integral of squared distance from $x$ to $T(x)$ when $T$ maps $p$ to $q$) :...
- 3,085
5
votes
Reference request: Long-term behaviour of the heat equation for bounded initial data
First of all, with initial data in $C_b$, classical solution exist, so there is no need for quotation marks. It is easy to see that the convolution of the initial data $f(x)$ with the Gauss–...
- 17.2k
5
votes
Heat equation with nonlocal boundary condition
A short observation, which is too long for a comment. Let's assume that $\Omega$ has unit measure, i.e. $|\Omega|=1$.
We define
$$
w = u - \int u \,\mathrm{d}x .
$$
By doing so, $w$ solves the ...
- 1,081
5
votes
Accepted
Hölder continuity in time of heat semigroup
No, such a bound is not possible.
We will show a counterexample for $d = 1$. For a given $T \in \mathbb{Z}_{+}$, consider the following function $\ell^{(T)} : \mathbb{R} \to \mathbb{R}_{\geq 0}$:
on ...
- 1,170
5
votes
Accepted
Hölder continuity in time of heat semigroup for regular initial distribution
I believe so.
Let $Z \sim \mathcal{N}(0,1)$.
Then
$$
\int_{\mathbb{R}^d} \mathrm{d}x \,(1 + |x|) \, \vert \ell_{t} (x) - \ell_{s} (x) \vert
= \int_{\mathbb{R}^d} \mathrm{d}x \,(1 + |x|) \, \left\vert ...
- 578
5
votes
Accepted
Existence of directional heat equation without uniform ellipticity
As you do not have any sort of coupling in any spatial direction other than $x_1$, what you have here is not actually a time-dependent PDE in $d$-dimensions but a $(d-1)$-parameter family of time-...
- 2,504
4
votes
Intuition for the Drift Term of the Laplace-Beltrami Operator
The usual way to think about the drift term is as the trace of the torsion tensor of the manifold. This interpretation is worked out, with analogies in hydrodynamics, by Diego Rapaport in [1] and [2].
- 188k
4
votes
Accepted
Singularity of the heat kernel
Yes, away from the boundary: the heat kernel for the interval is given by $$\tag{1}g(t,x,y)=(2\pi t)^{-1/2}\sum_{n\in\mathbb{Z}} (-1)^n \exp\left(-\frac{(x-y-n\pi)^2}{2t}\right),$$ and it is not ...
- 17.2k
4
votes
Accepted
heat kernel on closed manifolds - error in Chavel's book?
Yes, there is indeed a mistake. Chavels Lemma 2 on page 153 tells you that
$$L(H_k * F) = (LH_k)*F - F,$$
so if you define $F = \sum_{l=1}^\infty (LH_k)^{*l}$ and $p= H_k + H_k * F$, then
$$ L p = ...
- 9,853
4
votes
Gaussian distribution, maximum entropy and the heat equation
Here is yet another late answer, but I hope it is relevant.
Let me make first make clear that I use the mathematical "minus entropy" convention (as is common in my field, which is optimal transport). ...
- 5,380
4
votes
Decay time to constant function of heat kernel on 2-sphere
In the case of the standard metrics on the sphere, you can actually write everything explicitly using spherical harmonics. Namely, for each $l=0,1,2,\dots$, the Laplace-Beltrami operator on the sphere ...
- 8,992
4
votes
Accepted
Do eigenfunctions determine the geometry of a manifold? If so, do finitely many suffice?
Here is a sketch of an idea of how to show that the set $\mathcal{E}(g)\subset C^\infty(M)$ of all the eigenfunctions of the metric $g$ on a compact manifold $M$ determines $g$ up to a constant ...
- 108k
4
votes
Accepted
A question about positivity preserving property of semigroup of Laplacian
The first equality is known as the Trotter product formula. There are some hypotheses to check, and one should pay attention to the mode of convergence, but at a formal level, it's what you'd expect. ...
- 29.7k
Only top scored, non community-wiki answers of a minimum length are eligible