Hajer Bahouri, Isabelle Gallagher
The goal of this paper is to study the action of the heat operator on the Heisenberg group H^d, and in particular to characterize Besov spaces of negative index on H^d in terms of the heat kernel. That characterization can be extended to positive indexes using Bernstein inequalities. As a corollary we obtain a proof of refined Sobolev inequalities in W^{s,p} spaces.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The aim of this paper is threefold. First, we obtain the precise bounds for the heat kernel on isotropic Heisenberg groups by using well-known results in the three dimensional case. Second, we study the asymptotic estimates at infinity for the heat kernel on nonisotropic Heisenberg groups. As a consequence, we give uniform upper and lower estimates of the heat kernel, and complete its short-time behavior obtained by Beals-Gaveau-Greiner. Third, we complete the results obtained in \cite{Li12} about the heat kernel of Grushin operators.
In this note we obtain gradient estimates for the heat semigroup on nonisotropic Heisenberg groups. More precisely, our aim is to get the H.-Q. Li inequality on nonisotropic Heisenberg groups, which is a generalization of the original result on the classical Heisenberg group of dimension 3 and a counterpart of the inequality on the H-type groups. Our proof is based on a Cheeger type inequality, which is an approach proposed by Bakry et al.
About the convergence to initial data of the heat problem on the Heisenberg group
本文研究了Heisenberg群上热方程解的存在性，给出了初始数据f的可积条件，刻画了使得解在时间趋于零时几乎处处存在的加权Lebesgue空间，并得到了带权热核局部极大函数的有界性。
We study the following nonlocal diffusion equation in the Heisenberg group $\mathbb{H}_n$, \[ u_t(z,s,t)=J\ast u(z,s,t)-u(z,s,t), \] where $\ast$ denote convolution product and $J$ satisfies appropriated hypothesis. For the Cauchy problem we obtain that the asymptotic behavior of the solutions is the same form that the one for the heat equation in the Heisenberg group. To obtain this result we use the spherical transform related to the pair $(U(n),\mathbb{H}_n)$. Finally we prove that solutions of properly rescaled nonlocal Dirichlet problem converge uniformly to the solution of the correspond…
In this paper, we study a critical exponent to the semilinear heat equation with forcing term on Heisenberg group. Our technique of proof is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg group. Surprisingly, the critical exponent will be bounded for all dimensions of the Heisenberg group, in contrast to the Euclidean case which in the 1D and 2D cases will be infinite. In addition, we give an upper bound estimate on the lifespan of local solutions.
Let $L = -1/4 (\sum_{j=1}^n(X_j^2+Y_j^2)+iγT)$ where $γ$ is a complex number, $X_j$, $Y_j$, and $T$ are the left invariant vector fields of the Heisenberg group structure for $R^n \times R^n \times R$. We explicitly compute the Fourier transform (in the spatial variables) of the fundamental solution of the Heat Equation $\partial_sρ= -Lρ$. As a consequence, we have a simplified computation of the Fourier transform of the fundamental solution of the $\Box_b$-heat equation on the Heisenberg group and an explicit kernel of the heat equation associated to the weighted dbar-operator in $C^n$ with w…
In this paper we consider the semilinear Cauchy problem for the heat equation with power nonlinearity in the Heisenberg group $\mathbf{H}_n$. The heat operator is given in this case by $\partial_t-Δ_H$, where $Δ_H$ is the so-called sub-Laplacian on $\mathbf{H}_n$. We prove that the Fujita exponent $1 + 2/Q$ is critical, where $Q = 2n + 2$ is the homogeneous dimension of $\mathbf{H}_n$. Furthermore, we prove sharp lifespan estimates for local in time solutions in the subcritical case and in the critical case. In order to get the upper bound estimate for the lifespan (especially, in the critical…
We investigate observability and Lipschitz stability for the Heisenberg heat equation on the rectangular domain $$Ω= (-1,1)\times\mathbb{T}\times\mathbb{T}$$ taking as observation regions slices of the form $ω=(a,b) \times \mathbb{T} \times \mathbb{T}$ or tubes $ω= (a,b) \times ω_y \times \mathbb{T}$, with $-1<a<b<1$. We prove that observability fails for an arbitrary time $T>0$ but both observability and Lipschitz stability hold true after a positive minimal time, which depends on the distance between $ω$ and the boundary of $Ω$: $$T_{\min} \geqslant \frac{1}{8} \min\{(1+a)^2,(1-b)^2\}.$$ Our…
The Fujita exponent for a heat equation with mixed local and nonlocal nonlinearities on the Heisenberg group
本文研究了 Heisenberg 群上具混合局部与非局部非线性项的半线性热方程，通过容量方法确定了 Fujita 指数作为全局存在与有限时间爆破的分界线，并给出了超临界状态下的寿命估计。
In this paper we generalize the instantaneous blowup result from [3] and [15] to the heat equation perturbed by singular potentials on the Heisenberg group.
Heat content and horizontal mean curvature on the Heisenberg group
本文研究第一Heisenberg群中光滑有界区域的次Riemann热含量短时间渐近，利用Brown运动给出概率证明，并以水平周长和总水平平均曲率刻画其前几个系数。
Chernoff solutions of the heat and the Schrödinger equation in the Heisenberg group
该研究将经典Chernoff定理应用于海森堡群上的热方程与薛定谔方程，通过半群逼近得到基本解的解析与数值表示，建立热方程与布朗运动的新联系并严格实现费曼路径积分，为次黎曼流形偏微分方程研究开辟新方向。
Decay of mass for a semilinear heat equation on Heisenberg group
本文研究海森堡群上带时间依赖吸收的半线性热方程，证明非线性项与扩散效应在不同临界指数下决定大时间渐近行为。
Fujita exponent for the fractional sub-Laplace semilinear heat equation with forcing term on the Heisenberg group
本文在海森堡群上研究带强迫项的分数次亚拉普拉斯半线性热方程，确定其 Fujita 指数为 Q/(Q-2s)，并证明超临界情形全局解存在、次临界情形全局解不存在、临界情形存在有限时间爆破解，从而将经典 Fujita 现象推广到次黎曼非局部框架。
In this paper we use the heat equation in a group of Heisenberg type $\mathbb{G}$ to provide a unified treatment of the two very different extension problems for the time independent pseudo-differential operators $\mathscr L^s$ and $\mathscr L_s$, $0< s\leq 1$. Here, $\mathscr L^s$ is the fractional power of the horizontal Laplacian, and $\mathscr L_s$ is the conformal fractional power of the horizontal Laplacian on $\mathbb{G}$. One of our main objective is compute explicitly the fundamental solutions of these nonlocal operators by a new approach exclusively based on partial differential equa…
We study the behaviour of nonnegative solutions to the quasilinear heat equation with a reaction localized in a ball $$ u_t=Δu^m+a(x)u^p, $$ for $m>0$, $0<p\le\max\{1,m\}$, $a(x)=\mathds{1}_{B_L}(x)$, $0<L<\infty$ and $N\ge2$. We study when solutions, which are global in time, are bounded or unbounded. In particular we show that the precise value of the length $L$ plays a crucial role in the critical case $p=m$ for $N\ge3$. We also obtain the asymptotic behaviour of unbounded solutions and prove that the grow-up rate is different in most of the cases to the one obtained when $L=\infty$.
We study the behaviour of global solutions to the quasilinear heat equation with a reaction localized $$ u_t=(u^m)_{xx}+a(x) u^p, $$ $m, p>0$ and $a(x)$ being the characteristic function of an interval. we prove that there exists $p_0=\max\{1,\frac{m+1}2\}$ such that all global solution are bounded if $p>p_0$, while for $p\le p_0$ all the solution are global and unbounded. In the last case, we prove that if $p<m$ the grow-up rate is different to the one obtained when $a(x)\equiv1$, while if $p>m$ the grow-up rate coincides with that rate, but only inside the support of $a$; outside the interva…
By adapting a technique of Molchanov, we obtain the heat kernel asymptotics at the sub-Riemannian cut locus, when the cut points are reached by an $r$-dimensional parametric family of optimal geodesics. We apply these results to the bi-Heisenberg group, that is, a nilpotent left-invariant sub-Rieman\-nian structure on $\mathbb{R}^{5}$ depending on two real parameters $α_{1}$ and $α_{2}$. We develop some results about its geodesics and heat kernel associated to its sub-Laplacian and we illuminate some interesting geometric and analytic features appearing when one compares the isotropic ($α_{1}=…
We study logarithmic Sobolev inequalities with respect to a heat kernel measure on finite-dimensional and infinite-dimensional Heisenberg groups. Such a group is the simplest non-trivial example of a sub-Riemannian manifold. First we consider logarithmic Sobolev inequalities on non-isotropic Heisenberg groups. These inequalities are considered with respect to the hypoelliptic heat kernel measure, and we show that the logarithmic Sobolev constants can be chosen to be independent of the dimension of the underlying space. In this setting, a natural Laplacian is not an elliptic but a hypoelliptic …
We prove that for a general diffusion process, certain assumptions on its behavior \emph{only within a fixed open subset} of the state space imply the existence and sub-Gaussian type off-diagonal upper bounds of the \emph{global} heat kernel on the fixed open set. The proof is mostly probabilistic and is based on a seemingly new formula, which we call a \emph{multiple Dynkin-Hunt formula}, expressing the transition function of a Hunt process in terms of that of the part process on a given open subset. This result has an application to heat kernel analysis for the \emph{Liouville Brownian motio…
We study the heat kernel transform on a nilmanifold M associated to a H-type group. Using a reduction technique we reduce the problem to the case of Heisenberg groups. The image of $ L^2(M) $ under the heat kernel transform is shown to be a direct sum of weighted Bergman spaces.
This paper is part of a series papers devoted to geometric and spectral theoretic applications of the hypoelliptic calculus on Heisenberg manifolds. More specifically, in this paper we make use of the Heisenberg calculus of Beals-Greiner and Taylor to analyze the spectral theory of hypoelliptic operators on Heisenberg manifolds. The main results of this paper include: (i) Obtaining complex powers of hypoelliptic operators as holomorphic families of Psi_{H}DO's, which can be used to define a scale of weighted Sobolev spaces interpolating the weighted Sobolev spaces of Folland-Stein and providin…
Bounds on the logarithmic derivatives of the heat kernel on a compact Riemannian manifolds have been long known, and were recently extended, for the log-gradient and log-Hessian, to general complete Riemannian manifolds. Here, we further extend these bounds to incomplete Riemannan manifolds under the least restrictive condition on the distance to infinity available, for derivatives of all orders. Moreover, we consider not only the usual heat kernel associated to the Laplace-Beltrami operator, but we also allow the addition of a conservative vector field. We show that these bounds are sharp in …
Rapid stabilization of the heat equation with localized disturbance
本文提出了一种结合频率李雅普诺夫方法与符号多值算子的多值反馈控制策略，用于实现多维热方程在存在未知局部扰动下的快速稳定化，无需显式建模或估计扰动。
In this note we focus our attention on a stochastic heat equation defined on the Heisenberg group $\mathbf{H}^{n}$ of order $n$. This equation is written as $\partial_t u=\frac{1}{2}Δu+u\dot{W}_α$, where $Δ$ is the hypoelliptic Laplacian on $\mathbf{H}^{n}$ and $\{\dot{W}_α; α>0\}$ is a family of Gaussian space-time noises which are white in time and have a covariance structure generated by $(-Δ)^{-α}$ in space. Our aim is threefold: (i) Give a proper description of the noise $W_α$; (ii) Prove that one can solve the stochastic heat equation in the Itô sense as soon as $α>\frac{n}{2}$; (iii) Gi…
Exponential mixing for the stochastic Navier--Stokes equation with localized noise
本文针对带局域噪声的二维Navier-Stokes方程，结合Malliavin微积分与PDE控制理论，证明了相关Navier-Stokes-热系统的指数混合及速度律的指数收敛，为解决噪声退化问题提出了稳定转换方案。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。