Questions tagged [heat-equation]

The tag has no usage guidance.

231 questions
Filter by
Sorted by
Tagged with
98 views

What is the fundamental solution for the backward heat equation?

According to the theorem of Malgrange and Ehrenpreis a fundamental solution exists for any PDE with constant coefficients. But I didn't manage to find in the literature an explicit formula for the ...
• 2,675
125 views

Upper bound $\int_{\mathbb{R}^d \times \mathbb{R}^d} |fx)-f(y)| (1+|y|) \ell (x) p_t (x-y) \, \mathrm d x \, \mathrm d y$ in $t$

$\newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}}$ We fix $\alpha \in (0, 1)$ and $c>0$. Let $f : \bR^d \to \bR$ and $\ell : \bR^d \to \bR_+$ be measurable such that $\ell$ ...
• 1,085
72 views

Heat kernel convergence when expanding domains

Let $\Omega$ be an arbitrary domain in $\mathbb{R}^n$. There exists a positive $C^{\infty}$ function $G_{\Omega} : \Omega \times \Omega \times (0, \infty) \rightarrow \mathbb{R}$ (Dirichlet heat ...
• 625
136 views

Differential equation involving Casimir operator on $\operatorname{SL}(2,\mathbb{R})$

Throughout this post, we let $G$ denote the Lie group $\mathrm{SL}(2,\mathbb{R})$. For $t,\theta \in \mathbb{R}$ we define the following elements of $G$: \begin{align*} A(t) &= \begin{bmatrix}e^t &...
• 1,729
1 vote
61 views

Modulus of Continuity, Heat Flow, and Derivative Estimates

Given $f : \mathbf{R}^d \to \mathbf{R}$, define $P_t f$ by \begin{align} (P_t f)(x) = \mathbf{E} \left[ f (x + \sqrt{t} G) \right], \end{align} where $G \sim \mathcal{N} (0, I_d)$ is a standard ...
• 706
253 views

• 1,167
107 views

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 $\sigma : \mathbb ... • 1,085 1 vote 1 answer 117 views Is the heat kernel of a manifold$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)$is both integrable, and square-integrable ... • 5,353 0 votes 0 answers 58 views Existence of the asymptotic expansion of the heat kernel Theorem 2.26 in Heat Kernels and Dirac Operators says that the heat kernel of a Laplace type operator has a unique asymptotic expansion (the theorem and its proof are attached below). Unfortunately I ... • 329 1 vote 1 answer 61 views Upper bound$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}}, \... • 1,085 10 votes 0 answers 403 views 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 \... • 1,085 1 vote 1 answer 111 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. $$Let f : \mathbb R^d \to \... • 1,085 0 votes 0 answers 24 views Reference request heat equation with moving interface Let T,\sigma_1,\sigma_2>0, \lambda:[0,T]\to\mathbb{R} a continuous function and consider the following Cauchy problem on [0,T]\times \mathbb{R}:$$ \begin{cases} u_t = \sigma_1^2u_{xx} ~~~~\... • 431 0 votes 0 answers 95 views 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...
• 119
139 views

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 ...
• 31
137 views

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 ...
• 5,353
187 views

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 ...
• 5,203
201 views

• 1,085
284 views

• 11
1 vote
90 views

$L^2(0,\infty;L^2(\Omega))$ 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 ...
• 31
118 views

• 647
1 vote
89 views

• 647
1 vote
89 views

195 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 ...
• 655
363 views

• 1,055
1 vote
262 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 \...
• 2,875
53 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 ...
• 625
279 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/...
434 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 ...
• 53
254 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 ...
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 ...
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 \...