Questions tagged [heat-equation]

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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 ...
Bazin's user avatar
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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 ...
Luis Yanka Annalisc's user avatar
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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 ...
Akira's user avatar
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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 ...
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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 ...
Filippo's user avatar
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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}}, \...
Akira's user avatar
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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 \...
Akira's user avatar
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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 \...
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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} ~~~~\...
NancyBoy's user avatar
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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 $...
Snared's user avatar
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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 ...
Ciccisio's user avatar
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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 ...
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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 ...
leo monsaingeon's user avatar
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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)\...
NancyBoy's user avatar
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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|\...
Filippo's user avatar
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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 $$ ...
Akira's user avatar
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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 $\...
o0BlueBeast0o's user avatar
2 votes
0 answers
143 views

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) ...
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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) ...
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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 (\...
Akira's user avatar
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6 votes
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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 $C>0$ ...
mark's user avatar
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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 (\...
Abdelbadie Younes's user avatar
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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/...
Idempotent's user avatar
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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: $\sum_{\lvert \alpha\rvert=2k}\|D^{\alpha}u\|_{L^{2}(\Omega)}^{2}\leq |\Delta^{k}(u^{(l))}\|_{L^{2}(\Omega)}^{2}=\...
Amira's user avatar
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$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 ...
BBB's user avatar
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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-\...
Luis Yanka Annalisc's user avatar
1 vote
1 answer
58 views

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)...
Analyst's user avatar
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1 answer
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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)...
Analyst's user avatar
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3 votes
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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:...
thachung's user avatar
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 &...
Analyst's user avatar
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0 votes
1 answer
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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 $$ \...
Analyst's user avatar
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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 ...
Ilovemath's user avatar
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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 $\varepsilon>0$, ...
Guy Fsone's user avatar
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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\...
joaquindt's user avatar
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 ...
Student's user avatar
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1 answer
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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 ...
Bin Tang's user avatar
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)&...
Lorenzo Pompili's user avatar
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 \...
Ayman Moussa's user avatar
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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 ...
Ilovemath's user avatar
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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/...
Aleph1234's user avatar
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 ...
Adrien B's user avatar
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 ...
Fetchinson0234's user avatar
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 ...
Fetchinson0234's user avatar
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 \...
LL 3.14's user avatar
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0 answers
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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 ...
Falcon's user avatar
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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 ...
Antoine Balan's user avatar
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(\...
user484672's user avatar
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) = \...
Ilovemath's user avatar
  • 585
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 ...
joaquindt's user avatar
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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 $\frac{\partial u_l}{\partial t}-a^2\frac{\partial^2u_l}{\partial x^2}=f(x,t), (x,t)\in Q_T^l$...
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