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Questions tagged [heat-equation]

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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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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
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161 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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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
4 votes
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141 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
159 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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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
246 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
129 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
334 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
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120 views

Approximate solution of the heat equation

I'm reading the article "Singularity formation for the two dimensional harmonic map flow into $S^2$" from J.Davila, M.del Pino and J.Wei and at some point, there are some computations I don'...
Falcon's user avatar
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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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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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1 answer
127 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
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Fractional partial derivatives and integrals of De Bruijn's $H_t(z)$-function. Does a simpler form exist for the $z$ derivative/integral?

$\newcommand\KummerU{\text{KummerU}}$ $\newcommand\Hypergeom{\text{Hypergeom}}$ During 2018/2019, the polymath 15 project managed to successfully reduce the upper bound of the De Bruijn-Newman ...
Rudolph's user avatar
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0 answers
411 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
61 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
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2 votes
2 answers
114 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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47 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 $\frac{\partial u_l}{\partial t}-a^2\frac{\partial^2u_l}{\partial x^2}=f(x,t), (x,t)\in Q_T^l$...
xldd's user avatar
  • 119
3 votes
1 answer
233 views

Unique solutions to the heat equation on $\mathbb{R}^3$

Pierre-Gilles Lemarie-Rieusset, The Navier-Stokes Problem in the 21st Century treats the heat equation on $\mathbb{R}^3$ for time $t\geq 0$, and proves uniqueness of suitably smooth solutions by a ...
Colin McLarty's user avatar
2 votes
1 answer
216 views

What do higher order diffusion terms do?

I have been trying to learn to work with the Python module FiPy, which is supposed to solve PDEs of the form $$ \frac {\partial(\rho \phi)} {\partial t} - [\nabla\cdot(\Gamma_i\nabla)]^n\phi - \nabla \...
FusRoDah's user avatar
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2 votes
1 answer
178 views

Feynman-Kac formula with non-zero boundary condition

Let $D \subseteq \mathbb{R}^m$ be a bounded domain. The Feynman-Kac formula for the heat equation with initial condition $u(t, x) = f(x)$ and boundary condition $u(t, x)|_{\partial D} = 0$ is given by ...
user478954's user avatar
2 votes
0 answers
91 views

Gradient $L^\infty$-estimate for heat equation with homogeneous Dirichlet boundary condition

$\Omega\subset \mathbb{R}^N$ is a bounded smooth domain. Consider the homogeneous heat equation with zero boundary condition in $\Omega$ \begin{cases} \partial_t u-\Delta u=0 \quad(x,t)\in \Omega\...
Tibeku's user avatar
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1 vote
0 answers
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Where can I find a calculation of the asymptotic expansion of the heat kernel of the square of a Dirac operator?

Please note that I am not asking for a reference on asymptotic expansions in general (this was answered here), but rather for the concrete computation of the asymptotic expansion of a specific ...
Filippo's user avatar
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0 answers
95 views

$L^\infty-L^\infty$ bounds for heat semigroups constructed from the Dirichlet Laplacian

Let $D \subset \mathbb{R}^n$ be a bounded domain with Lipschitz boundary, and let $\Delta$ be the Laplace operator with the Dirichlet boundary condition on $D$. Let $e^{t\Delta}$ be the corresponding ...
SMS's user avatar
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1 vote
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61 views

Is there any way this property of semigroups can be satisfied?

Suppose you have the heat semigroup $(S(t))_{t>0}$, such that $$S(t)u(x) = (4\pi t)^{-n/2}\int_{\mathbb{R}^n}e^{-|x-y|^2/4t}u(y)dy.$$ The semigroup has the property that $$S(t)S(s)u(x) = S(t+s)u(x)....
Ilovemath's user avatar
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0 answers
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A question from Richard Hamilton's paper "A matrix Harnack estimate for the heat equation"

Richard Hamilton "A matrix Harnack estimate for the heat equation. Communications in Analysis and Geometry. 1(1993), 113-126." On page 125, at the end of the proof of Theorem 4.3, I abstract ...
Geom Zari's user avatar
4 votes
0 answers
85 views

$\mathcal{C}^1(\overline{\Omega})$ gradient bounds for the Dirichlet problem of the heat equation on general domains

I am studying the heat equation on a general bounded domain $\Omega \subset \mathbb{R}^+ \times \mathbb{R}^n$ with continuously differentiable Dirichlet data $\phi$ on the boundary, $$ \left\{ \begin{...
Theleb's user avatar
  • 173
1 vote
1 answer
173 views

Heat kernel on hyperbolic space of variable curvature

I am working with the heat kernel on the hyperbolic space explicitly (as you may guess by my previous questions) and I got the desired results when the curvature is $-\kappa=-1$. Now I am trying to do ...
MathqA's user avatar
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1 vote
1 answer
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Verify that a given function is a fundamental solution to the heat equation on the hyperbolic plane [closed]

A few days ago (see this), I asked a question regarding the derivatives of the function $$P_2(x,t)=\frac{\sqrt{2}e^{-t/4}}{(4\pi t)^{3/2}}\int_x^\infty\frac{se^{-s^2/4t}ds}{\sqrt{\cosh(s)-\cosh(x)}}.$$...
MathqA's user avatar
  • 163
1 vote
1 answer
217 views

Partial derivative of the heat kernel

I happen to have the heat kernel on the two-dimensional hyperbolic space and I need to take partial derivatives in order to check that it satisfies the heat equation as expected. The problem is I can ...
MathqA's user avatar
  • 163
2 votes
1 answer
234 views

Compactness for initial-to-final map for heat equation

Let $M$ be a compact smooth manifold without boundary. Let $T>0$ and let $g$ be a smooth Riemannian metric on $M$. Given any $f \in L^2(M)$ let $u$ be the unique solution to the equation $$\...
Ali's user avatar
  • 3,863
0 votes
0 answers
159 views

Has this form of the heat equation been solved for the radiation boundary condition

Below is a solution to a special form of the heat equation. I have found the postings on the heat equation and they are far above my head. I tried to find a tag on Transport Theory for both heat and ...
Michael Bernthal's user avatar
2 votes
0 answers
43 views

Covariance of an exclusion process

In Erhard and Hairer's recent paper, they say that the covariance of exclusion process is given by the discrete Heat Kernel (page 61, paragraph following equation 4.11). I have not been able to make ...
Usama's user avatar
  • 21
4 votes
0 answers
141 views

$L^\infty$ solutions for parabolic Neumann problem (heat equation)

Consider the heat equation on a (smooth) domain in $\mathbb{R}^n$ with homogeneous Neumann BCs: $$u_t - \Delta u = f$$ $$\partial_\nu u = 0$$ $$u|_{t=0} = u_0$$ where $f \in L^p(0,T;L^r(\Omega))$ and $...
soup's user avatar
  • 277
7 votes
0 answers
186 views

Li-Yau inequality on $\mathbb R^2$ for functions that are somewhat close to $1$

Let $u:\mathbb R^2\times \mathbb R_{>0}\to \mathbb R_{>0}$ be a positive solution to the heat equation on $\mathbb R^2$ ($u_{xx}+u_{yy}=u_t$, no constants). The Li-Yau inequality in this case ...
Alexander Kalmynin's user avatar
3 votes
0 answers
93 views

How to prove the existence of weak solutions of parabolic PDEs using Rothe's method?

I am reading a textbook on PDE (Introduction to Elliptic and Parabolic Partial Differential Equations by Z. Wu, J. Yin and C. Wang, written in Chinese) where a proof of the existence of weak solutions ...
Wentao Hu's user avatar
3 votes
0 answers
76 views

Ornstein-Ulhenbeck like semigroup for the orthogonal group with a hypercontractive inequality

I am looking for an Ornstein-Uhlenbeck like semigroup $P_t$ and associated generator $\mathcal{L}$ on $G = \operatorname{SO}(n)$ or $\operatorname{O}(n)$ that has a hypercontractive inequality with a ...
arjun's user avatar
  • 921
4 votes
0 answers
295 views

Estimates for the heat equation with inhomogeneous boundary condition

EDIT2: I believe the estimate required is false. There is some evidence that I have added to this post. I only believed it was true because it seemed like it was used in certain papers. However, in ...
Lorenzo Quarisa's user avatar
2 votes
1 answer
167 views

Lower Gaussian estimates for Dirichlet heat kernel on manifolds

Let $(M,g)$ be a Riemannian $n$-manifold with $Ric_g\ge -Kg$, $\Omega\subset M$ be an open subset. We can define Dirichlet heat kernel on $\Omega$, $p_{\Omega}(y,t,y',t')$ as the minimal fundamental ...
WhiteDwarf's user avatar
1 vote
0 answers
348 views

Martingales associated with heat equation

I am trying to learn the connection between Brownian motion and heat equation (in the spirit of Feynman-Kac, for example, here). I read (Michael E. Taylor's PDE book, Volume II, Chapter 11, ...
SMS's user avatar
  • 1,223
3 votes
3 answers
1k views

Uniqueness of solution to heat equation when initial condition is a generalized function

Let $u(t,x)$ be a solution to the heat equation $$\partial_t = \partial_{xx} \quad (t,x) \in [0,T) \times [-1,1]$$ subject to the initial/boundary conditions $$u(0,x) = f(x), \quad x \in [-1,1], \\ u(...
bm76's user avatar
  • 113
1 vote
0 answers
91 views

question about the book "Holomorphic Morse Inequalities" by Marinescu-Ma

Could somebody please explain to me the proof of proposition 1.6.4 (page 52) in the book "Holomorphic Morse Inequalities" by Marinescu-Ma? I am completely lost. One point that is really ...
Andres Larrain-Hubach's user avatar
2 votes
0 answers
47 views

Mixed boundary value problems for Heat equation

This might be a very simple question, but basically I am looking for a good reference for studying the heat equation on Riemannian manifolds with boundary, specifically when data is put on lateral ...
Ali's user avatar
  • 3,863
3 votes
1 answer
77 views

How to solve a differential equation in the form $\frac{\partial}{\partial t}g(x,t)=g(x-\Delta,t)+\frac{\partial^2}{\partial x^2} g(x,t)$?

How to find the general solution of a differential equation with a shift, in the following form? $$\frac{\partial}{\partial t}g(x,t)=g(x-\Delta,t)+\frac{\partial^2}{\partial x^2} g(x,t)$$ where $\...
user1611107's user avatar
2 votes
0 answers
94 views

Laplacian coupled with another equation over a two-dimensional rectangular region

I have the two-dimensional Laplacian $(\nabla^2 T(x,y)=0)$ coupled with another equation which is: $$\frac{\partial t}{\partial x}+\alpha(t-T)=0 \tag 1$$ where it is known that $t(x=0)=t_i$. The ...
Avrana's user avatar
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5 votes
0 answers
294 views

heat kernel Asymptotic expansion on manifolds with boundary or manifolds with conical singularities

This is a similar question to Heat Kernel Asymptotics on Manifold with Boundary. But I have some further questions. Let $(M,g)$ be a closed Riemannian manifold, the heat kernel $p(x,y,t)$ of Laplace-...
WhiteDwarf's user avatar