Questions tagged [heat-equation]
The heat-equation tag has no usage guidance.
226
questions
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Non-selfadjoint operators and physical systems
There are plenty of examples of non-selfadjoint operators modelizing physical phenomena: to name a few, let's quote the the heat equation (\ref{HEAT}, see below), the Navier-Stokes system for ...
2
votes
0
answers
88
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Harmonic heat flow, formal and rigorous
Let $ (M,g) $ be a smooth Riemann manifold without boundary, $ S^{n-1} $ is an $ n $-dimensional sphere, and $ T>0 $. Consider a weak solution $ u:M\times[0,T]\to S^{n+1} $ of
$$
\partial_tu-\Delta ...
2
votes
0
answers
102
views
Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$
Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it.
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb ...
1
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1
answer
95
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Is the heat kernel of a manifold $p$-integrable?
If $M$ is a separable, oriented Riemannian manifold, without any other assumption on its geometry, and $h$ is its heat kernel, it is known that $h(t,x,\cdot)$ is both integrable, and square-integrable ...
0
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48
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Existence of the asymptotic expansion of the heat kernel
Theorem 2.26 in Heat Kernels and Dirac Operators says that the heat kernel of a Laplace type operator has a unique asymptotic expansion (the theorem and its proof are attached below). Unfortunately I ...
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1
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58
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Upper bound $I_R := \int_{B_R^c} |x| (P_t \ell_\nu) (x) \, \mathrm d x$ in terms of $R, \nu, t$?
Let $(p_t)_{t >0}$ be the Gaussian heat kernel on $\mathbb R^d$ and $(P_t)_{t >0}$ its induced semi-group, i.e.,
$$
\begin{align}
p_t (x) &:= (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\...
10
votes
0
answers
387
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Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (u (t, x))|^2 u(t, x) \}$
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R \to \mathbb R$ belong to the Hölder space $C^{1, \alpha}_b (\mathbb R)$ for some $\alpha \in (0, 1)$. Let $u : \mathbb T \times \...
1
vote
1
answer
109
views
An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) \partial_1 \partial_1 g_t (x-y) \, \mathrm d y$
Let $(g_t)_{t>0}$ be the Gaussian heat kernel on $\mathbb R^d$, i.e.,
$$
g_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d.
$$
Let $f : \mathbb R^d \to \...
0
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0
answers
22
views
Reference request heat equation with moving interface
Let $T,\sigma_1,\sigma_2>0$, $\lambda:[0,T]\to\mathbb{R}$ a continuous function and consider the following Cauchy problem on $[0,T]\times \mathbb{R}$:
$$
\begin{cases}
u_t = \sigma_1^2u_{xx} ~~~~\...
0
votes
0
answers
80
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Reversing heat transfer with respect to time
Fact: One can easily compute heat dispersion in a plane using the heat equation.
Question: Has any research been done on computing the process in the reverse time direction?
That is, given a heat map $...
2
votes
1
answer
96
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Gradient flows: evolution of geodesics
I’m trying to understand if, when I move the marginals of a Wasserstein geodesic along a contractive flow, the geodesic between the new probability measures is “near” to the geodesic
connecting the ...
2
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0
answers
127
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The existence of a positive Green function for the Laplacian on $\mathbb R$
One can show explicitly and easily that the function $G(x,y) = \frac 1 2 |x-y|$ is a positive Green function for the Laplacian $\frac {\mathrm d ^2} {\mathrm d x ^2}$ on $\mathbb R$ (endowed with the ...
3
votes
2
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167
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Heat equation with nonlocal boundary condition
$\newcommand{\R}{\mathbb R}$I am looking for any possible references (modelling, mathematical and numerical analysis, well-posedness, regularity, anything really!) for heat equation-like problems with ...
2
votes
1
answer
187
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Bound for zero-crossings of heat equation
I am considering the following problem.
Let $\mathcal{P}$ the classical heat-diffusion problem:
$$\mathcal{P} : \left(\partial_t u (t,x)=\frac{1}{2}\partial_{xx}^2u(t,x)\text{ with }u(0,\cdot) = f(x)\...
2
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0
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127
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Heat kernels and Dirac operators - Why are half densities invoked in the definition of heat kernels?
The authors of Heat kernels and Dirac operators chose to consider a generalized Laplacian $H$ on a bundle $E\otimes|\Lambda|^{1/2}$ and heat kernels of $H$ are defined to be sections of $(E\otimes|\...
4
votes
1
answer
738
views
How to get $\int_{\mathbb R^d} |\partial_i\partial_j(1-\Delta)^{-\frac{\delta}{2}}p_t(\cdot-y)(x)| \, \mathrm d x \lesssim t^{\frac{\delta}{2}-1}$?
We consider the Gaussian heat kernel $p_t$ on $\mathbb R^d$, i.e.,
$$
p_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d,
$$
and define the operator $P_t$ by
$$
...
3
votes
1
answer
274
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Heat kernel of left-invariant metric on 3-sphere
This post concerns the heat equation on $S^3 (\simeq \mathrm{SU}(2))$ endowed with an arbitrary left-invariant metric. We think of $S^3$ as the space of unit quaternions, and its Lie algebra $\...
2
votes
0
answers
143
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Positivity of the Fourier transform: prove or disprove that $\operatorname{Re}(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\xi))\geq0$
Let $F:[0,\infty) \to[0,\infty)$ be increasing, $C^1$ and $L-$Lipschitz with $F(0)=0$. Let $u\in L^1 (\Bbb R^d)$, $u\geq0$ so that $F\circ u\in L^1 (\Bbb R^d)$
I would like to prove (or disprove) ...
1
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0
answers
94
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Scaling limit of a discrete analogue of the heat equation
For $f \in L^1 (\mathbb R^d)$, given $\varepsilon > 0$, define the function $T_\varepsilon f$ on $\mathbb R^d$ by
$$T_\varepsilon f(x) := \frac{1}{|B_\varepsilon (x)|} \int_{B_\varepsilon (x)} f(y) ...
0
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1
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93
views
Does $c$ in the embedding inequality $\|P^\kappa_t \|_{L^p \to L^{p'}} \le c t^{-\frac{d(p'-p)}{2pp'}}$ depend on $\kappa$?
For any $\kappa>0$, we consider the Gaussian heat kernel
$$
p^\kappa_t (x) := (\kappa \pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{\kappa t}},
\quad t>0, x \in {\mathbb R}^d.
$$
Let $L^0 := L^0 (\...
6
votes
0
answers
147
views
Gaussian lower heat kernel bounds on non-convex bounded domain
I am looking for a proof the following theorem.
Let $U \subset \mathbb{R}^n$ be a bounded domain with $C^2$ boundary and $p(x,y,t)$ be the Neumann heat kernel. Then there exist a constant $C>0$ ...
1
vote
0
answers
89
views
Gradient estimate of Dirichlet Heat kernel (Classical Laplacian)
Let $p^D(t,x,y)$ be the heat kernel for the Dirichlet Laplacian in an open set $D$. Do we have the following estimate and where can I find it ? $$\lvert\nabla_xp^D(t,x,y)\rvert\le C\dfrac{1}{\min (\...
0
votes
0
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28
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Is an asymptotic expansion of the heat kernel known on any space of fractal dimension?
Is an asymptotic expansion of the heat kernel known on any space of fractal dimension? I was able to find a 2012 paper saying it was not (https://www.math.tugraz.at/~grabner/Publications/...
1
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0
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94
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Regularity of solution to heat equation
If $u$ is solution of $u_{t}=\Delta u$ in a bounded domain $\Omega$, is it true that:
$\sum_{\lvert \alpha\rvert=2k}\|D^{\alpha}u\|_{L^{2}(\Omega)}^{2}\leq |\Delta^{k}(u^{(l))}\|_{L^{2}(\Omega)}^{2}=\...
1
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0
answers
85
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$L^2(0,\infty;L^2(\Omega))$ estimate on solution of heat equation with Neumann boundary condition
Let $u$ be a solution of
$$u' - \Delta u = 0 \quad\text{on $\Omega$}$$
$$\partial_\nu u = 0\quad\text{in $\partial \Omega$}$$
$$u(t=0)=u_0\quad\text{on $\Omega$}$$
where $\Omega$ is a bounded ...
2
votes
0
answers
110
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Local smoothness of harmonic heat flow
Assume that $ (M,g) $ is a smooth closed manifold and $ \mathbb{S}^{L-1} $ is a unit sphere with dimension $ L-1 $ in $ \mathbb{R}^{L} $. Consider the equation of harmonic heat flow
$$
\partial_tu-\...
1
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1
answer
58
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Are there $f,h$ such that $h$ is Lipschitz, $\int_0^t f(s)\,\mathrm d s<\infty$ and $|\partial_t g| (t, x) \le f(t)g(h(t), x)$?
We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R,\quad (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
Then
$$
\partial_t g(t, x)...
1
vote
1
answer
85
views
Is there $f$ such that $\int_0^t f(s)\,\mathrm d s<\infty$ and $|\partial_t g| (t, x) \le f(t)g(Ct, x)$ for all $t>0$ and $x \in \mathbb R^d$?
We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R,\quad (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
Then
$$
\partial_t g(t, x)...
3
votes
0
answers
120
views
Number of spatial critical points of a solution to the heat equation in higher dimensions
I would like to know if the number of spatial critical points of a solution to the heat equation can increase. Given $u_0:\mathbb S^n\to\mathbb R$, let $u$ be the solution of the initial value problem:...
3
votes
1
answer
81
views
Let $g$ be the heat kernel. Are there constants $C_1, C_2>0$ such that $\frac{g(t_1, \cdot)}{t_1} \le C_1 \frac{g(C_2 t_2, \cdot)}{\sqrt{t_2}}$?
We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R,\quad (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
Let $0 < t_1 < t_2 &...
0
votes
1
answer
163
views
An estimate of the gradient of heat kernel
We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R, (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
I have already proved that
$$
\...
1
vote
0
answers
78
views
Non-existence of classical solutions of Hardy PDE
On the paper "On the Cauchy Problem for Reaction-Diffusion Equations" Wang studies the Hardy-Hénon equation
$$
\begin{cases}
u_t - \Delta u = |\cdot|^{l}u^{p}& \mbox{ in } \mathbb{R}^n ...
1
vote
0
answers
52
views
Comparison principle for porous medium equation in Fourier variables
Let $V:[0,\infty) \to[0,\infty)$ be convex, $C^2$ with $V(0)=0$. Define $F(u): = uV'(u)-V(u) $. Let $v\in L^1 (\Bbb R^d)$, $v\geq0$ so that $F\circ v\in L^1 (\Bbb R^d)$. For fixed $\varepsilon>0$, ...
1
vote
0
answers
158
views
Regularity up to the boundary of solutions of the heat equation
Given the heat problem:
$$\begin{cases}
\frac{d}{dt}u(x,t)=\Delta u(x,t) & \forall (x,t)\in \Omega\times(0,T) \\
u(x,0)=u_0(x) & \forall x\in\Omega \\
u(x,t)=0 & \forall x\in\partial\...
5
votes
1
answer
190
views
Convergence of heat flow on non-compact manifolds?
Consider the heat equation $\partial_t u= \Delta u+\lambda_1 u$ on a non-compact complete manifold $M$ (with nonpositive curvature) where $\lambda_1$ is the first eigenvalue and we start with some ...
5
votes
1
answer
345
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The decay rate of a degenerate heat equation in torus $\mathbb{T}^2$
Consider the degenerate heat equation on torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$:
$$ \frac{\partial}{\partial t} u(x,t)= \left( \sin^2(\pi x_1) \frac{\partial^2}{\partial^2 x_2} + \sin^2(\pi ...
5
votes
0
answers
308
views
All $L^pL^q$ estimates for the heat equation on $\mathbb R$ (with gain of derivatives)
I have asked this question on MSE, but this is a better place.
The heat equation and the heat kernel.
Consider the heat equation on $\mathbb R$:
$$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&...
1
vote
0
answers
233
views
Maximal regularity heat equation
Considering the heat equation on the flat torus $\mathbf{T}^d$, we have the maximal regularity estimate
\begin{align*}
\forall \varphi\in\mathscr{D}(\mathbf{R}\times\mathbf{T}^d),\quad \|\Delta \...
2
votes
0
answers
52
views
Are there results that relate the restriction of the heat semigroup in $\mathbb{R}^n$ and the semigroup in a domain?
I'm thinking about the following situation:0 suppose that
$$
S_{\Omega}(t)f = \int_{\Omega} K_{\Omega}(x,y,t)f(y)dy
$$ where $K_\Omega(x,y ,t)$ is the Dirichlet heat kernel in the domain $\Omega$. It ...
2
votes
1
answer
264
views
Relationship between heat kernel and Maxwell-Boltzmann distribution
There appears to be a connection between the heat kernel and Maxwell-Boltzmann distribution, but I have not seen this in the literature before. I'd appreciate any kind comments or corrections/...
5
votes
1
answer
417
views
Ricci flow negative curvature
We denote by $\mathbb{H}^n$ the hyperbolic plane of dimension $n$.
I don't know much about the Ricci flow, so my first question is probably naïve : can one define a normalized Ricci flow such that the ...
2
votes
1
answer
225
views
Heat conduction type equation in 4D
[I asked a similar question, Linear PDE, analytic continuation, Green's function and boundary conditions, and was told that a follow-up question should be a separate post.]
I'm interested in a ...
3
votes
1
answer
374
views
Linear PDE, analytic continuation, Green's function and boundary conditions
I'm looking at the linear PDE in 3+1 dimensions,
$$
\left[ -(\partial_t - \xi)^2 - \partial_k \partial_k \right] \phi(t,x) = 4\pi^2 \delta(t)\delta(x)\label{1} \tag{1}
$$
Where $\xi$ is generally a ...
4
votes
0
answers
141
views
Approximation by gaussian mollification in Sobolev spaces
I have been trying to find a good estimate on the constant in the inequality (in dimension $d=3$ to simplify)
$$\label{0}\tag{0}
\|(1-e^{t\Delta})f\|_{L^{5/3}(\Bbb R^3)} \leq C_0\,t^{2/5}\, \|\nabla \...
2
votes
0
answers
102
views
Noether's theorem in the critical heat equation
I initially posted this question on Stackexchange Mathematics (see here) but as I got no answer, maybe someone here will be able to help me.
I am watching a serie of lectures on "Blow up solution ...
2
votes
1
answer
201
views
The heat equation for complex time
Let $\Delta$ be a Laplacian or an elliptic operator over a manifold, can the heat equation be defined for complex time? Can we define:
$$e^{-z \Delta}$$
for $Re(z)>0$ ?
Also can the Ricci flow be ...
1
vote
0
answers
438
views
Heat kernel on quaternion Heisenberg group
For the n Heisenberg($\Bbb C^n\times\Bbb R$) it is known that the heat kernel $q_s(z,t)=c_n\int_{\Bbb R} e^{-i\lambda t}\Big( \frac{\lambda}{\sinh(\lambda s)}\Big)^n e^{-\frac{\lambda|z|^2\coth(\...
2
votes
0
answers
96
views
Representation of heat kernel in general domains
I'm looking for results on the representation of the heat kernel in general domains. If $e^{-\Delta_{\Omega} t}$ denotes the heat semigroup in $\Omega$, we have to
$$ (e^{-\Delta_{\Omega} t}f)(x) = \...
3
votes
2
answers
194
views
Sharp Dirichlet heat kernel estimates in exterior domains?
I am right now working on some linear parabolic problems studying the behaviour of its solutions for large initial data. To do this, I have needed to use some estimates of the Dirichlet and Neumann ...
0
votes
0
answers
54
views
increasing property of the heat equation on the interval
$Q_T^l=\{(x,t); 0<x<l, 0<t\leq T\}$,
$u_l\in C^{2,1}(Q_T^l)\cap C(\bar Q_T^l)$ is the solution of $\frac{\partial u_l}{\partial t}-a^2\frac{\partial^2u_l}{\partial x^2}=f(x,t), (x,t)\in Q_T^l$...